Case Study 3

Michael Daniel Bigler and Liam Arthur Phan

2023-05-03

Packages

First we load all the necessary packages for the analysis.

library(psych)
library(corrplot)
library(ggplot2)
library(car)
library(naniar)
library(REdaS)
library(zoo)
library(foreign) 
library(lavaan)
library(lavaanPlot)
library(ggcorrplot)
library(lares)
library(MVN)
library(dplyr)
library(knitr)
library(dplyr)
library(kableExtra)
library(semPlot)

Data

Here we load the data and select only the necessary value for the analysis.

df <- read.csv2('Case Study III_Structural Equation Modeling.csv', na.strings = '999', sep = ',')
df <- df[, c(1:23, 25:36)]

DT::datatable(df)

Dimensions

To see how much data we have we look at the dimensions of the data.

dim_before_na <- dim(df)
dim_before_na
## [1] 553  35

We see that we have 553 rows and 35 columns.

Summary Statistics

Here we use summary statistics on the data.

DT::datatable(describe(df))

Missing Analysis

As not all further analysis do work with missing values we need to check the existence of them.

gg_miss_var(df, show_pct = TRUE)

We see that there are a lot of columns which have some percentage of values missing. To see if the missing values are in some observations only or if they are spread out between a lot of the observations we use a plot which highlights the missing values in every observation.

naniar::vis_miss(df)

If we look at this plot though we see that the missing values are in a lot of the observations. Most EFA methods do not work well with missing values. For the Confirmatory Analysis we will need to choose a method which can handle missing values.

Dimensions after list wise deletion

Based on the input of the assistant we will use list wise deletion for EFA.

dim_after_na <- dim(na.omit(df))
dim_after_na
## [1] 309  35
na_remove_count <- dim_after_na - dim_before_na
na_remove_count[1] <- abs(na_remove_count[1])

Thus, we remove 244 with list wise deletion which is a lot of observations compared to the size of the whole dataset.

# We do list-wise deletion as asked by the TA
df_listwise <- na.omit(df)

Assumptions for EFA

From Assistant Please only consider variables image1 to image22, and use list wise deletion to handle missing data before starting exploratory factor analysis.

Basic Assumptions

Here we select the images which whe need for the EFA.

df_1 <- df_listwise[,1:22]

Normality - Shapiro Wilk’s test

In this part we check if the data is normally distributed.

apply(df_1, 2, shapiro.test)
## $Im1
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92373, p-value = 1.851e-11
## 
## 
## $Im2
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92499, p-value = 2.411e-11
## 
## 
## $Im3
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92371, p-value = 1.844e-11
## 
## 
## $Im4
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92014, p-value = 8.873e-12
## 
## 
## $Im5
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.91233, p-value = 1.921e-12
## 
## 
## $Im6
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.82674, p-value < 2.2e-16
## 
## 
## $Im7
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.84612, p-value < 2.2e-16
## 
## 
## $Im8
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.80379, p-value < 2.2e-16
## 
## 
## $Im9
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92158, p-value = 1.187e-11
## 
## 
## $Im10
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.7981, p-value < 2.2e-16
## 
## 
## $Im11
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.85448, p-value < 2.2e-16
## 
## 
## $Im12
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.86819, p-value = 1.38e-15
## 
## 
## $Im13
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.89122, p-value = 4.669e-14
## 
## 
## $Im14
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.79446, p-value < 2.2e-16
## 
## 
## $Im15
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.91767, p-value = 5.414e-12
## 
## 
## $Im16
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90488, p-value = 4.853e-13
## 
## 
## $Im17
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90676, p-value = 6.818e-13
## 
## 
## $Im18
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.93189, p-value = 1.081e-10
## 
## 
## $Im19
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90393, p-value = 4.097e-13
## 
## 
## $Im20
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92824, p-value = 4.834e-11
## 
## 
## $Im21
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.89126, p-value = 4.7e-14
## 
## 
## $Im22
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.94596, p-value = 3.211e-09

We reject the null-hypothesis for all variables and thus don’t accept normality of the data.

Multivariate normality - Mardia’s Multivariate Normality Test

To say the data are multivariate normal:

• z-kurtosis < 5 (Bentler, 2006) and the P-value should be ≥ 0.05. • The plot should also form a straight line (Arifin, 2015).

MVN::mvn(df_1, mvnTest = "mardia", multivariatePlot = "qq", desc = FALSE)

## $multivariateNormality
##              Test        Statistic p value Result
## 1 Mardia Skewness 5693.58260956909       0     NO
## 2 Mardia Kurtosis 48.5688465675536       0     NO
## 3             MVN             <NA>    <NA>     NO
## 
## $univariateNormality
##                Test  Variable Statistic   p value Normality
## 1  Anderson-Darling    Im1       9.2497  <0.001      NO    
## 2  Anderson-Darling    Im2       9.0183  <0.001      NO    
## 3  Anderson-Darling    Im3       9.0339  <0.001      NO    
## 4  Anderson-Darling    Im4       9.6661  <0.001      NO    
## 5  Anderson-Darling    Im5      10.8951  <0.001      NO    
## 6  Anderson-Darling    Im6      18.6917  <0.001      NO    
## 7  Anderson-Darling    Im7      17.3319  <0.001      NO    
## 8  Anderson-Darling    Im8      21.2033  <0.001      NO    
## 9  Anderson-Darling    Im9       9.2733  <0.001      NO    
## 10 Anderson-Darling   Im10      22.1406  <0.001      NO    
## 11 Anderson-Darling   Im11      16.4825  <0.001      NO    
## 12 Anderson-Darling   Im12      14.7775  <0.001      NO    
## 13 Anderson-Darling   Im13      12.4484  <0.001      NO    
## 14 Anderson-Darling   Im14      21.7494  <0.001      NO    
## 15 Anderson-Darling   Im15      10.1132  <0.001      NO    
## 16 Anderson-Darling   Im16      11.4400  <0.001      NO    
## 17 Anderson-Darling   Im17      10.7737  <0.001      NO    
## 18 Anderson-Darling   Im18       8.0021  <0.001      NO    
## 19 Anderson-Darling   Im19      12.4287  <0.001      NO    
## 20 Anderson-Darling   Im20       7.9905  <0.001      NO    
## 21 Anderson-Darling   Im21      13.1862  <0.001      NO    
## 22 Anderson-Darling   Im22       6.1723  <0.001      NO

The data are not normally distributed at multivariate level. Our extraction method PAF can deal with this non-normality.

Multicolinearity

Here we check for multicolinearity in the data. First of all we will do a correlation plot and then a correlation ranking plot.

# Correlation Values Matrix
M <- cor(df_1)

# P-Value
p.mat <- cor_pmat(df_1)
Correlation Plot

Here we see the correlation plot of the images.

# Correlation Plot
ggcorrplot(M, hc.order = TRUE, type = "lower", lab = TRUE, p.mat = p.mat, sig.level=0.05, lab_size = 2, tl.cex = 10,outline.col = "white", ggtheme = ggplot2::theme_minimal(), colors = c("#823038", "white", "#2596be")) 

We see that there are some correlations between the images.

Correlation Ranking

Here we show the highest correlation from the plot before.

# Ranked Cross-Correlations
corr_cross(df_1, # name of dataset
  max_pvalue = 0.05, # display only significant correlations (at 5% level)
  top = 9 # display top 10 couples of variables (by correlation coefficient)
)

As we can see, We have some multicolinearity amongst the variables, at least 6 variables can be considered with high-colinearity. Im3+Im4, Im1+Im2, Im6+Im7, Im4+Im5, Im8+Im10 and Im8+Im14.

A guide to appropriate use of Correlation coefficient in medical research

Factors Analysis Assumptions

Here we check the assumptions needed for factor analysis.

Kaiser-Meyer-Olkin test (KMO)

First of all we look at the kaiser-meyer-olkin test which checks the sampling adequacy of all of our images. Thereby a score classifies the adequacy as it can be seen in the picture below.

KMO: Find the Kaiser, Meyer, Olkin Measure of Sampling Adequacy

KMO Index

Now we show the KMO value for all of our images.

KMOTEST <- KMO(M)
sort(KMOTEST$MSAi)
##       Im6      Im10      Im14       Im2       Im1       Im7      Im20      Im17 
## 0.7791619 0.8192843 0.8206186 0.8275596 0.8316756 0.8342908 0.8369658 0.8459668 
##      Im18       Im4       Im3      Im13      Im12      Im22      Im21      Im11 
## 0.8479170 0.8623498 0.8647696 0.8749019 0.8763560 0.8850423 0.8930068 0.9101259 
##      Im16       Im8       Im9      Im19      Im15       Im5 
## 0.9168866 0.9231784 0.9240378 0.9432565 0.9558911 0.9616355

As one can see most KMO Index are either Meritorious or Marvelous. Im6 is the lowest KMO index being Middling.

KMO Overall Measure of sampling adequacy

From all of these scores a general sampling adequacy can be computed.

KMOTEST$MSA
## [1] 0.8739058

With 0.87 the overall sampling adequacy is very high.

Bartlett’s Test of Sphericity

This test checks if there are correlations between the variables as EFA can not be done if there wouldn’t be.

cortest.bartlett(df_1)
## $chisq
## [1] 5268.134
## 
## $p.value
## [1] 0
## 
## $df
## [1] 231

The test says that EFA can be done as the test indicates a p-value of 0 (P-value = 0) and thus we can reject the null hypothesis that the correlation matrix look like the identity matrix.

Exploratory Factor analysis

Now we do an exploratory factor analysis of our images.

Determine the number of factors

To determine the number of factors there are five different criterions one can use which are listed below.

  1. Kaiser’s eigenvalue > 1 rule.
  2. Cattell’s scree test.
  3. Parallel analysis.
  4. Very simple structure (VSS).
  5. Velicer’s minimum average partial (MAP).

Kaiser’s eigevalue > 1 rule

Factors with eigenvalues > 1 are retained. Eigenvalue can be interpreted as the proportion of the information in a factor. The cut-off of 1 means the factor contains information = 1 item. Thus it is not worthwhile keeping factor with information < 1 item.

fa_result <- fa(df_1, rotate = "varimax", fm = "pa")

factors_kaiser <- sum(fa_result$e.values>1)

According to the Kaiser-Criterion, we would use 6 factors.

Catell’s scree test

For the scree test criterion one needs to look at the plot of the initial eigenvalues against the used factors and choose the value where there is a dent in the curve. Here we did a factor analysis using rotation varimax and plotted the initial eigenvalues

fa_result <- fa(df_1, rotate = "varimax", fm = "pa")
n_factors <- length(fa_result$e.values)
scree <- data.frame(Factor_n =  as.factor(1:n_factors), Eigenvalue = fa_result$e.values)

ggplot(scree, aes(x = Factor_n, y = Eigenvalue, group = 1)) +
  geom_point() + geom_line() +
  xlab("Number of factors") +
  ylab("Initial eigenvalue") +
  labs( title = "Scree Plot",
        subtitle = "(Based on the unreduced correlation matrix)") +
  geom_hline(yintercept = 1, color="#2596be") + theme_minimal() 

As one can see from the plot the number of factors to choose would either be 7 or 8. Additionally in the scree plot one can see the kaiser criterion which selects all the factors above the blue line of eigenvalue > 1.

Parallel analysis

Here we do a parallel factor analyis.

parallel <- fa.parallel(df_1, fm = "pa", fa = "fa")

## Parallel analysis suggests that the number of factors =  6  and the number of components =  NA
print(parallel)
## Call: fa.parallel(x = df_1, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors =  6  and the number of components =  NA 
## 
##  Eigen Values of 
## 
##  eigen values of factors
##  [1]  8.52  1.78  0.92  0.76  0.65  0.58  0.20  0.13 -0.09 -0.12 -0.21 -0.25
## [13] -0.29 -0.34 -0.37 -0.40 -0.42 -0.42 -0.47 -0.51 -0.54 -0.60
## 
##  eigen values of simulated factors
##  [1]  0.62  0.45  0.37  0.32  0.27  0.23  0.18  0.14  0.10  0.06  0.02 -0.01
## [13] -0.05 -0.09 -0.11 -0.15 -0.19 -0.23 -0.26 -0.30 -0.35 -0.39
## 
##  eigen values of components 
##  [1] 9.11 2.46 1.58 1.36 1.26 1.14 0.79 0.73 0.56 0.46 0.36 0.33 0.30 0.28 0.25
## [16] 0.22 0.19 0.18 0.14 0.11 0.10 0.08
## 
##  eigen values of simulated components
## [1] NA

As we can see in parallel analysis, it also suggest 6 factors, nevertheless, factors up to 7 or 8 can also be considered.

Very simple structure (VSS) criterion and Velicer’s minimum average partial (MAP) criterion

Here we do the VSS and MAP criterion.

vss(df_1, rotate = "varimax", fm = "pa")

## 
## Very Simple Structure
## Call: vss(x = df_1, rotate = "varimax", fm = "pa")
## VSS complexity 1 achieves a maximimum of 0.84  with  1  factors
## VSS complexity 2 achieves a maximimum of 0.9  with  2  factors
## 
## The Velicer MAP achieves a minimum of 0.04  with  8  factors 
## BIC achieves a minimum of  -334.88  with  8  factors
## Sample Size adjusted BIC achieves a minimum of  -71.64  with  8  factors
## 
## Statistics by number of factors 
##   vss1 vss2   map dof chisq     prob sqresid  fit RMSEA  BIC SABIC complex
## 1 0.84 0.00 0.047 209  2756  0.0e+00    16.0 0.84 0.199 1558  2221     1.0
## 2 0.71 0.90 0.044 188  2143  0.0e+00    10.1 0.90 0.183 1065  1661     1.3
## 3 0.62 0.89 0.045 168  1773 1.0e-265     7.6 0.92 0.176  810  1343     1.4
## 4 0.52 0.84 0.047 149  1455 8.0e-213     5.9 0.94 0.168  600  1073     1.7
## 5 0.44 0.76 0.044 131  1139 3.0e-160     4.3 0.96 0.158  388   803     1.9
## 6 0.39 0.63 0.042 114   636  1.9e-73     3.0 0.97 0.122  -18   344     2.1
## 7 0.39 0.60 0.045  98   296  1.1e-21     2.3 0.98 0.081 -266    45     1.9
## 8 0.37 0.54 0.036  83   141  7.5e-05     1.7 0.98 0.047 -335   -72     2.0
##   eChisq  SRMR eCRMS eBIC
## 1   2321 0.127 0.134 1122
## 2   1213 0.092 0.102  135
## 3    837 0.077 0.090 -126
## 4    588 0.064 0.080 -266
## 5    364 0.051 0.067 -387
## 6    157 0.033 0.047 -497
## 7     74 0.023 0.035 -488
## 8     19 0.012 0.019 -456

VSS indicates 1 or 2 factors (vss1 largest at 1 and 2 factors), while MAP indicates 8 factors (map smallest at 8 factors).

VSS criterion for the number of factors (in R’s psych package)

Extraction Method

Our data are not normally distributed, hence the extraction method of choice is principal axis factoring (PAF), because it does not assume normality of data (Brown, 2015). The rotation method is varimax.

We run EFA by

  1. fixing the number of factors as decided from previous step. 6 or 8 factors are reasonable.
  2. choosing an appropriate extraction method. We use PAF, fm = “pa” (Principal Axis Factoring).
  3. choosing an appropriate rotation method. We use varimax, rotate = “varimax”.

6 Factors

We will compute the loadings with 6 factors and varimax rotation

What we need to look for:

  1. Factor loadings

Multiple threshold exist (as many rules of thumb), in our analysis we will use the standard 0.4 cut-off.

What thresholds should I use for factor loading cut-offs?

  1. Communalities

We use the standard cut-off of 0.5, all above are good.

fa_result <- fa(df_1, nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1, nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA5   PA1    PA2   PA4   PA3   PA6    h2    u2  com
## Im1  0.850                                0.841 0.159 1.34
## Im2  0.842                                0.796 0.204 1.25
## Im3        0.831                          0.874 0.126 1.58
## Im4        0.850                          0.893 0.107 1.51
## Im5        0.603                          0.524 0.476 1.98
## Im6                                 0.783 0.708 0.292 1.31
## Im7               0.445             0.739 0.767 0.233 1.75
## Im8               0.725                   0.716 0.284 1.72
## Im9                                 0.480 0.453 0.547 2.80
## Im10              0.821                   0.793 0.207 1.37
## Im11                    0.537             0.461 0.539 2.24
## Im12                    0.796             0.758 0.242 1.42
## Im13                    0.748             0.725 0.275 1.64
## Im14              0.794                   0.760 0.240 1.43
## Im15 0.589                                0.641 0.359 2.86
## Im16 0.468                                0.461 0.539 3.00
## Im17                    0.439       0.416 0.666 0.334 4.39
## Im18                    0.438             0.605 0.395 4.18
## Im19 0.478 0.405                          0.541 0.459 3.38
## Im20                          0.839       0.775 0.225 1.21
## Im21                          0.766       0.680 0.320 1.34
## Im22                          0.798       0.804 0.196 1.56
## 
##                         PA5   PA1   PA2   PA4   PA3   PA6
## SS loadings           2.864 2.782 2.608 2.514 2.395 2.080
## Proportion Var        0.130 0.126 0.119 0.114 0.109 0.095
## Cumulative Var        0.130 0.257 0.375 0.489 0.598 0.693
## Proportion Explained  0.188 0.183 0.171 0.165 0.157 0.136
## Cumulative Proportion 0.188 0.370 0.541 0.706 0.864 1.000
## 
## Mean item complexity =  2.1
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  231  with the objective function =  17.57 with Chi Square =  5268.134
## df of  the model are 114  and the objective function was  2.15 
## 
## The root mean square of the residuals (RMSR) is  0.033 
## The df corrected root mean square of the residuals is  0.047 
## 
## The harmonic n.obs is  309 with the empirical chi square  157.019  with prob <  0.00471 
## The total n.obs was  309  with Likelihood Chi Square =  635.908  with prob <  1.9e-73 
## 
## Tucker Lewis Index of factoring reliability =  0.7871
## RMSEA index =  0.1217  and the 90 % confidence intervals are  0.1128 0.1312
## BIC =  -17.693
## Fit based upon off diagonal values = 0.993
## Measures of factor score adequacy             
##                                                     PA5   PA1   PA2   PA4   PA3
## Correlation of (regression) scores with factors   0.933 0.949 0.930 0.906 0.930
## Multiple R square of scores with factors          0.871 0.900 0.864 0.821 0.865
## Minimum correlation of possible factor scores     0.741 0.800 0.729 0.643 0.731
##                                                     PA6
## Correlation of (regression) scores with factors   0.904
## Multiple R square of scores with factors          0.817
## Minimum correlation of possible factor scores     0.635

Exploratory factor analysis and Cronbach’s alpha Questionnaire Validation Workshop, 10/10/2017, USM Health Campus

1. Factor loadings

We can see that we have 3 cross-loadings, Im7, Im17 and Im19.

Cross-Loadings (Measured with Complexity measure: com > 1):

Im17 > Im19 > Im7 > 1

2. Communalities

On the table, it is column h2

Low Communalities are :

Im9 < Im11 < Im16 < 0.5

Removing Im17 (Lowest Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17")], nfactors = 6, rotate = "varimax", 
##     fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA2   PA3   PA4   PA6    h2     u2  com
## Im1  0.855                               0.848 0.1519 1.33
## Im2  0.845                               0.799 0.2007 1.25
## Im3        0.839                         0.884 0.1159 1.55
## Im4        0.871                         0.923 0.0769 1.46
## Im5        0.605                         0.526 0.4743 1.97
## Im6                                0.860 0.795 0.2050 1.15
## Im7                                0.792 0.786 0.2142 1.51
## Im8              0.673             0.429 0.692 0.3077 1.99
## Im9                                0.471 0.449 0.5511 2.83
## Im10             0.873                   0.878 0.1222 1.32
## Im11                         0.548       0.458 0.5423 2.13
## Im12                         0.835       0.805 0.1950 1.33
## Im13                         0.757       0.738 0.2618 1.63
## Im14             0.830                   0.818 0.1820 1.40
## Im15 0.593                               0.643 0.3570 2.80
## Im16 0.468                               0.461 0.5390 3.00
## Im18                                     0.421 0.5791 4.38
## Im19 0.480 0.403                         0.538 0.4623 3.36
## Im20                   0.836             0.770 0.2296 1.21
## Im21                   0.773             0.684 0.3157 1.30
## Im22                   0.800             0.804 0.1959 1.55
## 
##                         PA1   PA5   PA2   PA3   PA4   PA6
## SS loadings           2.778 2.690 2.447 2.382 2.367 2.057
## Proportion Var        0.132 0.128 0.117 0.113 0.113 0.098
## Cumulative Var        0.132 0.260 0.377 0.490 0.603 0.701
## Proportion Explained  0.189 0.183 0.166 0.162 0.161 0.140
## Cumulative Proportion 0.189 0.371 0.538 0.699 0.860 1.000
## 
## Mean item complexity =  1.9
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  210  with the objective function =  16.063 with Chi Square =  4821.666
## df of  the model are 99  and the objective function was  1.019 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.036 
## 
## The harmonic n.obs is  309 with the empirical chi square  81.338  with prob <  0.902 
## The total n.obs was  309  with Likelihood Chi Square =  301.667  with prob <  2.4e-22 
## 
## Tucker Lewis Index of factoring reliability =  0.9055
## RMSEA index =  0.0813  and the 90 % confidence intervals are  0.0711 0.0921
## BIC =  -265.934
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA2   PA3   PA4
## Correlation of (regression) scores with factors   0.935 0.959 0.943 0.931 0.915
## Multiple R square of scores with factors          0.875 0.920 0.889 0.866 0.837
## Minimum correlation of possible factor scores     0.750 0.841 0.777 0.733 0.673
##                                                     PA6
## Correlation of (regression) scores with factors   0.924
## Multiple R square of scores with factors          0.854
## Minimum correlation of possible factor scores     0.709

Removing Im18 (Lowest loadings and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18")], nfactors = 6, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA2   PA3   PA4   PA6    h2     u2  com
## Im1  0.859                               0.853 0.1475 1.32
## Im2  0.846                               0.798 0.2016 1.24
## Im3        0.844                         0.895 0.1053 1.55
## Im4        0.873                         0.928 0.0722 1.47
## Im5        0.599                         0.520 0.4797 2.00
## Im6                                0.859 0.787 0.2125 1.14
## Im7                                0.810 0.799 0.2007 1.44
## Im8              0.652             0.448 0.685 0.3146 2.12
## Im9                                0.461 0.427 0.5731 2.87
## Im10             0.877                   0.890 0.1099 1.33
## Im11                         0.549       0.458 0.5420 2.14
## Im12                         0.853       0.835 0.1654 1.31
## Im13                         0.743       0.723 0.2772 1.67
## Im14             0.829                   0.823 0.1771 1.42
## Im15 0.597                               0.644 0.3563 2.76
## Im16 0.470                               0.460 0.5402 2.98
## Im19 0.482                               0.535 0.4651 3.33
## Im20                   0.838             0.772 0.2276 1.20
## Im21                   0.773             0.681 0.3192 1.29
## Im22                   0.802             0.805 0.1952 1.54
## 
##                         PA1   PA5   PA2   PA3   PA4   PA6
## SS loadings           2.738 2.565 2.379 2.353 2.223 2.059
## Proportion Var        0.137 0.128 0.119 0.118 0.111 0.103
## Cumulative Var        0.137 0.265 0.384 0.502 0.613 0.716
## Proportion Explained  0.191 0.179 0.166 0.164 0.155 0.144
## Cumulative Proportion 0.191 0.370 0.537 0.701 0.856 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.509 with Chi Square =  4660.488
## df of  the model are 85  and the objective function was  0.921 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.037 
## 
## The harmonic n.obs is  309 with the empirical chi square  72.686  with prob <  0.827 
## The total n.obs was  309  with Likelihood Chi Square =  273.088  with prob <  1.36e-21 
## 
## Tucker Lewis Index of factoring reliability =  0.9046
## RMSEA index =  0.0846  and the 90 % confidence intervals are  0.0736 0.0961
## BIC =  -214.246
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA2   PA3   PA4
## Correlation of (regression) scores with factors   0.937 0.963 0.946 0.931 0.921
## Multiple R square of scores with factors          0.877 0.927 0.894 0.868 0.847
## Minimum correlation of possible factor scores     0.755 0.855 0.789 0.735 0.695
##                                                     PA6
## Correlation of (regression) scores with factors   0.926
## Multiple R square of scores with factors          0.858
## Minimum correlation of possible factor scores     0.717

Removing Im8 (Cross-loadings (High Complexity))

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8")], nfactors = 6, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA3   PA4   PA2   PA6    h2     u2  com
## Im1  0.859                               0.853 0.1471 1.32
## Im2  0.844                               0.796 0.2045 1.24
## Im3        0.845                         0.893 0.1070 1.54
## Im4        0.875                         0.928 0.0721 1.45
## Im5        0.598                         0.519 0.4808 2.00
## Im6                          0.884       0.817 0.1827 1.10
## Im7                          0.806       0.768 0.2316 1.38
## Im9                          0.464       0.427 0.5730 2.80
## Im10                               0.882 0.924 0.0763 1.39
## Im11                   0.553             0.455 0.5449 2.08
## Im12                   0.861             0.841 0.1592 1.28
## Im13                   0.740             0.717 0.2833 1.67
## Im14                               0.811 0.824 0.1761 1.54
## Im15 0.599                               0.645 0.3550 2.72
## Im16 0.472                               0.454 0.5461 2.89
## Im19 0.483 0.405                         0.534 0.4664 3.27
## Im20             0.836                   0.770 0.2303 1.21
## Im21             0.774                   0.682 0.3183 1.29
## Im22             0.803                   0.807 0.1933 1.54
## 
##                         PA1   PA5   PA3   PA4   PA2   PA6
## SS loadings           2.730 2.578 2.347 2.243 1.942 1.812
## Proportion Var        0.144 0.136 0.124 0.118 0.102 0.095
## Cumulative Var        0.144 0.279 0.403 0.521 0.623 0.719
## Proportion Explained  0.200 0.189 0.172 0.164 0.142 0.133
## Cumulative Proportion 0.200 0.389 0.561 0.725 0.867 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  171  with the objective function =  14.415 with Chi Square =  4336.563
## df of  the model are 72  and the objective function was  0.819 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.039 
## 
## The harmonic n.obs is  309 with the empirical chi square  67.363  with prob <  0.633 
## The total n.obs was  309  with Likelihood Chi Square =  243.141  with prob <  2.01e-20 
## 
## Tucker Lewis Index of factoring reliability =  0.9011
## RMSEA index =  0.0876  and the 90 % confidence intervals are  0.0758 0.1001
## BIC =  -169.66
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA3   PA4   PA2
## Correlation of (regression) scores with factors   0.936 0.963 0.931 0.923 0.930
## Multiple R square of scores with factors          0.877 0.928 0.867 0.852 0.866
## Minimum correlation of possible factor scores     0.754 0.855 0.735 0.703 0.731
##                                                     PA6
## Correlation of (regression) scores with factors   0.953
## Multiple R square of scores with factors          0.908
## Minimum correlation of possible factor scores     0.816

Removing Im19 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8","Im19")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8", "Im19")], 
##     nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA5   PA4   PA3   PA1   PA2   PA6    h2     u2  com
## Im1        0.876                         0.894 0.1059 1.34
## Im2        0.863                         0.839 0.1610 1.26
## Im3  0.843                               0.886 0.1137 1.53
## Im4  0.884                               0.938 0.0620 1.43
## Im5  0.615                               0.538 0.4624 1.93
## Im6                          0.886       0.820 0.1796 1.09
## Im7                          0.805       0.765 0.2346 1.37
## Im9                          0.464       0.428 0.5717 2.79
## Im10                               0.892 0.937 0.0628 1.38
## Im11                   0.553             0.454 0.5462 2.06
## Im12                   0.866             0.843 0.1568 1.26
## Im13                   0.743             0.715 0.2854 1.64
## Im14                               0.812 0.822 0.1779 1.53
## Im15       0.560                         0.623 0.3767 3.06
## Im16       0.405                         0.396 0.6043 3.39
## Im20             0.844                   0.779 0.2213 1.19
## Im21             0.773                   0.679 0.3214 1.28
## Im22             0.801                   0.805 0.1948 1.54
## 
##                         PA5   PA4   PA3   PA1   PA2   PA6
## SS loadings           2.460 2.387 2.343 2.248 1.932 1.792
## Proportion Var        0.137 0.133 0.130 0.125 0.107 0.100
## Cumulative Var        0.137 0.269 0.399 0.524 0.632 0.731
## Proportion Explained  0.187 0.181 0.178 0.171 0.147 0.136
## Cumulative Proportion 0.187 0.368 0.546 0.717 0.864 1.000
## 
## Mean item complexity =  1.7
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  153  with the objective function =  13.52 with Chi Square =  4071.795
## df of  the model are 60  and the objective function was  0.48 
## 
## The root mean square of the residuals (RMSR) is  0.018 
## The df corrected root mean square of the residuals is  0.029 
## 
## The harmonic n.obs is  309 with the empirical chi square  32.204  with prob <  0.999 
## The total n.obs was  309  with Likelihood Chi Square =  142.579  with prob <  1.12e-08 
## 
## Tucker Lewis Index of factoring reliability =  0.9455
## RMSEA index =  0.0667  and the 90 % confidence intervals are  0.0528 0.0811
## BIC =  -201.422
## Fit based upon off diagonal values = 0.998
## Measures of factor score adequacy             
##                                                     PA5   PA4   PA3   PA1   PA2
## Correlation of (regression) scores with factors   0.966 0.950 0.933 0.925 0.931
## Multiple R square of scores with factors          0.933 0.902 0.870 0.856 0.867
## Minimum correlation of possible factor scores     0.867 0.804 0.740 0.712 0.734
##                                                     PA6
## Correlation of (regression) scores with factors   0.958
## Multiple R square of scores with factors          0.919
## Minimum correlation of possible factor scores     0.837

6 Factors - Conclusion

We removed Im17, Im18, Im8 and Im9 until achieving clear loadings separation.

fa.diagram(fa_result, sort = TRUE, adj = 1, rsize = 4, e.size = 0.07, main = "Factors Analysis with 6 factors", digits = 2, l.cex = 1)

Most Factors have good loadings (at least 2 above 0.7), while PA6 has only 2 variables loaded.

8 Factors

We will redo the same analysis with 8 factors this time and using varimax rotation as well.

fa_result <- fa(df_1, nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1, nfactors = 8, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA2   PA6   PA7   PA5    PA4   PA8    h2     u2  com
## Im1                                0.879              0.949 0.0505 1.49
## Im2                                0.822              0.832 0.1679 1.50
## Im3        0.803                                      0.877 0.1232 1.81
## Im4        0.858                                      0.939 0.0606 1.61
## Im5        0.620                                      0.568 0.4319 2.09
## Im6                          0.845                    0.773 0.2268 1.17
## Im7                          0.819                    0.808 0.1924 1.42
## Im8              0.619       0.474                    0.700 0.2995 2.45
## Im9                          0.447                    0.444 0.5557 3.44
## Im10             0.870                                0.887 0.1126 1.37
## Im11                   0.552                          0.467 0.5327 2.18
## Im12                   0.851                          0.841 0.1588 1.35
## Im13                   0.724                          0.730 0.2703 1.88
## Im14             0.850                                0.860 0.1396 1.41
## Im15                               0.442        0.423 0.668 0.3322 4.90
## Im16                                            0.732 0.744 0.2559 1.87
## Im17                                      0.830       0.923 0.0768 1.77
## Im18                                      0.757       0.785 0.2147 1.83
## Im19                                            0.561 0.640 0.3605 3.44
## Im20 0.849                                            0.790 0.2095 1.20
## Im21 0.769                                            0.684 0.3161 1.33
## Im22 0.797                                            0.799 0.2012 1.56
## 
##                         PA3   PA1   PA2   PA6   PA7   PA5   PA4   PA8
## SS loadings           2.441 2.394 2.308 2.244 2.110 2.100 1.718 1.395
## Proportion Var        0.111 0.109 0.105 0.102 0.096 0.095 0.078 0.063
## Cumulative Var        0.111 0.220 0.325 0.427 0.523 0.618 0.696 0.760
## Proportion Explained  0.146 0.143 0.138 0.134 0.126 0.126 0.103 0.083
## Cumulative Proportion 0.146 0.289 0.427 0.562 0.688 0.814 0.917 1.000
## 
## Mean item complexity =  2
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  231  with the objective function =  17.57 with Chi Square =  5268.134
## df of  the model are 83  and the objective function was  0.479 
## 
## The root mean square of the residuals (RMSR) is  0.012 
## The df corrected root mean square of the residuals is  0.019 
## 
## The harmonic n.obs is  309 with the empirical chi square  19.471  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  140.985  with prob <  7.46e-05 
## 
## Tucker Lewis Index of factoring reliability =  0.9674
## RMSEA index =  0.0474  and the 90 % confidence intervals are  0.0337 0.0609
## BIC =  -334.883
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA2   PA6   PA7
## Correlation of (regression) scores with factors   0.934 0.958 0.947 0.920 0.924
## Multiple R square of scores with factors          0.872 0.917 0.898 0.846 0.854
## Minimum correlation of possible factor scores     0.745 0.834 0.795 0.692 0.709
##                                                     PA5   PA4   PA8
## Correlation of (regression) scores with factors   0.964 0.943 0.838
## Multiple R square of scores with factors          0.930 0.890 0.703
## Minimum correlation of possible factor scores     0.860 0.780 0.405

1. Factor loadings

We can see that we have 2 cross-loadings, Im8 and Im15. Therefore 1 less cross-loadings than 6 Factors Analysis.

Cross-Loadings (Measured with Complexity measure: com > 1):

Im15 > Im8 > 1

2. Communalities

On the table, it is column h2

Low Communalities are :

Im9 < Im11 < 0.5 (same low items communalities than in 6 Factor analysis )

Removing Im15 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im15")], nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15")], nfactors = 8, rotate = "varimax", 
##     fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA2   PA5   PA7   PA6    PA4   PA8    h2     u2  com
## Im1                                0.864              0.935 0.0647 1.55
## Im2                                0.825              0.845 0.1553 1.52
## Im3        0.804                                      0.878 0.1225 1.80
## Im4        0.856                                      0.936 0.0639 1.61
## Im5        0.625                                      0.573 0.4275 2.06
## Im6                          0.850                    0.781 0.2190 1.16
## Im7                          0.817                    0.805 0.1952 1.42
## Im8              0.623       0.472                    0.698 0.3019 2.41
## Im9                          0.444                    0.442 0.5578 3.46
## Im10             0.881                                0.902 0.0979 1.35
## Im11                   0.556                          0.471 0.5294 2.14
## Im12                   0.852                          0.841 0.1587 1.34
## Im13                   0.722                          0.725 0.2749 1.88
## Im14             0.837                                0.844 0.1564 1.43
## Im16                                            0.675 0.677 0.3234 2.11
## Im17                                      0.826       0.912 0.0883 1.75
## Im18                                      0.765       0.794 0.2060 1.80
## Im19                                            0.609 0.686 0.3136 3.02
## Im20 0.854                                            0.796 0.2037 1.19
## Im21 0.770                                            0.683 0.3169 1.32
## Im22 0.796                                            0.795 0.2051 1.55
## 
##                         PA3   PA1   PA2   PA5   PA7   PA6   PA4   PA8
## SS loadings           2.395 2.369 2.303 2.152 2.086 1.841 1.699 1.174
## Proportion Var        0.114 0.113 0.110 0.102 0.099 0.088 0.081 0.056
## Cumulative Var        0.114 0.227 0.336 0.439 0.538 0.626 0.707 0.763
## Proportion Explained  0.150 0.148 0.144 0.134 0.130 0.115 0.106 0.073
## Cumulative Proportion 0.150 0.297 0.441 0.575 0.706 0.821 0.927 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  210  with the objective function =  16.523 with Chi Square =  4959.748
## df of  the model are 70  and the objective function was  0.434 
## 
## The root mean square of the residuals (RMSR) is  0.012 
## The df corrected root mean square of the residuals is  0.02 
## 
## The harmonic n.obs is  309 with the empirical chi square  17.262  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  127.971  with prob <  2.9e-05 
## 
## Tucker Lewis Index of factoring reliability =  0.9627
## RMSEA index =  0.0517  and the 90 % confidence intervals are  0.0374 0.0659
## BIC =  -273.363
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA2   PA5   PA7
## Correlation of (regression) scores with factors   0.935 0.954 0.949 0.919 0.925
## Multiple R square of scores with factors          0.873 0.911 0.900 0.844 0.856
## Minimum correlation of possible factor scores     0.747 0.821 0.800 0.688 0.712
##                                                     PA6   PA4   PA8
## Correlation of (regression) scores with factors   0.957 0.938 0.806
## Multiple R square of scores with factors          0.917 0.881 0.649
## Minimum correlation of possible factor scores     0.834 0.761 0.299

Removing Im8 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im15","Im8")], nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15", "Im8")], nfactors = 8, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA4   PA5   PA6   PA2   PA7   PA8    h2     u2  com
## Im1                          0.898                   0.990 0.0105 1.49
## Im2                          0.792                   0.800 0.1997 1.61
## Im3        0.800                                     0.875 0.1250 1.82
## Im4        0.855                                     0.937 0.0632 1.62
## Im5        0.627                                     0.575 0.4248 2.06
## Im6                    0.885                         0.827 0.1732 1.12
## Im7                    0.804                         0.766 0.2335 1.39
## Im9                    0.443                         0.440 0.5599 3.42
## Im10                               0.914             0.980 0.0201 1.37
## Im11             0.562                               0.468 0.5320 2.07
## Im12             0.857                               0.845 0.1553 1.32
## Im13             0.722                               0.723 0.2773 1.86
## Im14                               0.780             0.789 0.2110 1.64
## Im16                                           0.616 0.601 0.3986 2.35
## Im17                                     0.864       0.969 0.0311 1.66
## Im18                                     0.730       0.752 0.2484 1.92
## Im19                                           0.689 0.758 0.2415 2.40
## Im20 0.852                                           0.794 0.2064 1.19
## Im21 0.770                                           0.683 0.3173 1.32
## Im22 0.798                                           0.798 0.2024 1.55
## 
##                         PA3   PA1   PA4   PA5   PA6   PA2   PA7   PA8
## SS loadings           2.384 2.345 2.169 1.936 1.832 1.788 1.698 1.216
## Proportion Var        0.119 0.117 0.108 0.097 0.092 0.089 0.085 0.061
## Cumulative Var        0.119 0.236 0.345 0.442 0.533 0.623 0.708 0.768
## Proportion Explained  0.155 0.153 0.141 0.126 0.119 0.116 0.110 0.079
## Cumulative Proportion 0.155 0.308 0.449 0.575 0.694 0.810 0.921 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.431 with Chi Square =  4636.932
## df of  the model are 58  and the objective function was  0.36 
## 
## The root mean square of the residuals (RMSR) is  0.011 
## The df corrected root mean square of the residuals is  0.021 
## 
## The harmonic n.obs is  309 with the empirical chi square  15.22  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  106.281  with prob <  0.000114 
## 
## Tucker Lewis Index of factoring reliability =  0.9638
## RMSEA index =  0.0518  and the 90 % confidence intervals are  0.036 0.0674
## BIC =  -226.253
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA4   PA5   PA6
## Correlation of (regression) scores with factors   0.934 0.955 0.922 0.931 0.989
## Multiple R square of scores with factors          0.873 0.911 0.850 0.867 0.978
## Minimum correlation of possible factor scores     0.746 0.823 0.700 0.735 0.955
##                                                     PA2   PA7   PA8
## Correlation of (regression) scores with factors   0.980 0.973 0.818
## Multiple R square of scores with factors          0.961 0.947 0.669
## Minimum correlation of possible factor scores     0.922 0.895 0.338

8 Factors - Conclusion

We removed Im15, Im8 until achieving clear loadings separation. Therefore we removed 2 variables less than 6 Factors Analysis done previously

fa.diagram(fa_result, sort = TRUE, adj = 1, rsize = 4, e.size = 0.07, main = "Factors Analysis with 8 factors", digits = 2, l.cex = 1)

Most Factors have nice loadings (at least 2 above 0.7), but PA8 has 2 variables with only 0.62-0.69 loadings (but close to 0.7).

Deciding between 6 or 8 Factors

Here we show the results of the final factor analysis with 6 and 8 Factors and then evaluate how we decided between the two solutions.

fa_result6 <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8","Im19")], nfactors = 6, fm = "pa", rotate = "varimax")
fa_result8 <- fa(df_1[!names(df_1) %in% c("Im15","Im8")], nfactors = 8, fm = "pa", rotate = "varimax")

fa_result6
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8", "Im19")], 
##     nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##       PA5  PA4  PA3  PA1  PA2  PA6   h2    u2 com
## Im1  0.19 0.88 0.22 0.18 0.07 0.08 0.89 0.106 1.3
## Im2  0.22 0.86 0.16 0.10 0.07 0.08 0.84 0.161 1.3
## Im3  0.84 0.21 0.22 0.22 0.12 0.14 0.89 0.114 1.5
## Im4  0.88 0.21 0.18 0.20 0.14 0.13 0.94 0.062 1.4
## Im5  0.62 0.24 0.18 0.20 0.08 0.15 0.54 0.462 1.9
## Im6  0.09 0.06 0.07 0.06 0.89 0.12 0.82 0.180 1.1
## Im7  0.06 0.08 0.11 0.11 0.80 0.29 0.77 0.235 1.4
## Im9  0.19 0.13 0.12 0.36 0.46 0.11 0.43 0.572 2.8
## Im10 0.19 0.10 0.05 0.21 0.23 0.89 0.94 0.063 1.4
## Im11 0.18 0.11 0.18 0.55 0.09 0.25 0.45 0.546 2.1
## Im12 0.17 0.15 0.10 0.87 0.10 0.16 0.84 0.157 1.3
## Im13 0.22 0.24 0.19 0.74 0.14 0.07 0.71 0.285 1.6
## Im14 0.17 0.14 0.06 0.20 0.27 0.81 0.82 0.178 1.5
## Im15 0.26 0.56 0.26 0.37 0.17 0.10 0.62 0.377 3.1
## Im16 0.35 0.40 0.13 0.20 0.06 0.22 0.40 0.604 3.4
## Im20 0.14 0.11 0.84 0.17 0.03 0.04 0.78 0.221 1.2
## Im21 0.16 0.18 0.77 0.12 0.08 0.04 0.68 0.321 1.3
## Im22 0.21 0.24 0.80 0.14 0.19 0.06 0.81 0.195 1.5
## 
##                        PA5  PA4  PA3  PA1  PA2  PA6
## SS loadings           2.46 2.39 2.34 2.25 1.93 1.79
## Proportion Var        0.14 0.13 0.13 0.12 0.11 0.10
## Cumulative Var        0.14 0.27 0.40 0.52 0.63 0.73
## Proportion Explained  0.19 0.18 0.18 0.17 0.15 0.14
## Cumulative Proportion 0.19 0.37 0.55 0.72 0.86 1.00
## 
## Mean item complexity =  1.7
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  153  with the objective function =  13.52 with Chi Square =  4071.79
## df of  the model are 60  and the objective function was  0.48 
## 
## The root mean square of the residuals (RMSR) is  0.02 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic n.obs is  309 with the empirical chi square  32.2  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  142.58  with prob <  1.1e-08 
## 
## Tucker Lewis Index of factoring reliability =  0.946
## RMSEA index =  0.067  and the 90 % confidence intervals are  0.053 0.081
## BIC =  -201.42
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    PA5  PA4  PA3  PA1  PA2  PA6
## Correlation of (regression) scores with factors   0.97 0.95 0.93 0.93 0.93 0.96
## Multiple R square of scores with factors          0.93 0.90 0.87 0.86 0.87 0.92
## Minimum correlation of possible factor scores     0.87 0.80 0.74 0.71 0.73 0.84
fa_result8
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15", "Im8")], nfactors = 8, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##       PA3  PA1  PA4  PA5  PA6  PA2  PA7  PA8   h2    u2 com
## Im1  0.23 0.18 0.19 0.07 0.90 0.08 0.13 0.18 0.99 0.010 1.5
## Im2  0.18 0.21 0.11 0.07 0.79 0.08 0.17 0.21 0.80 0.200 1.6
## Im3  0.22 0.80 0.20 0.11 0.14 0.13 0.17 0.26 0.88 0.125 1.8
## Im4  0.18 0.86 0.20 0.13 0.15 0.13 0.16 0.23 0.94 0.063 1.6
## Im5  0.18 0.63 0.19 0.06 0.21 0.17 0.18 0.07 0.58 0.425 2.1
## Im6  0.07 0.08 0.05 0.88 0.04 0.12 0.11 0.05 0.83 0.173 1.1
## Im7  0.11 0.06 0.11 0.80 0.06 0.28 0.07 0.06 0.77 0.234 1.4
## Im9  0.12 0.17 0.33 0.44 0.08 0.12 0.26 0.06 0.44 0.560 3.4
## Im10 0.05 0.16 0.20 0.22 0.05 0.91 0.05 0.15 0.98 0.020 1.4
## Im11 0.18 0.18 0.56 0.09 0.09 0.25 0.08 0.06 0.47 0.532 2.1
## Im12 0.10 0.15 0.86 0.09 0.09 0.14 0.14 0.15 0.84 0.155 1.3
## Im13 0.18 0.19 0.72 0.12 0.18 0.08 0.26 0.11 0.72 0.277 1.9
## Im14 0.06 0.16 0.20 0.28 0.11 0.78 0.05 0.14 0.79 0.211 1.6
## Im16 0.12 0.26 0.15 0.05 0.24 0.19 0.15 0.62 0.60 0.399 2.3
## Im17 0.20 0.20 0.22 0.17 0.19 0.06 0.86 0.17 0.97 0.031 1.7
## Im18 0.19 0.24 0.24 0.16 0.14 0.06 0.73 0.13 0.75 0.248 1.9
## Im19 0.16 0.28 0.20 0.14 0.25 0.16 0.18 0.69 0.76 0.242 2.4
## Im20 0.85 0.12 0.17 0.03 0.07 0.03 0.07 0.12 0.79 0.206 1.2
## Im21 0.77 0.15 0.11 0.08 0.15 0.04 0.15 0.06 0.68 0.317 1.3
## Im22 0.80 0.20 0.13 0.18 0.20 0.07 0.15 0.07 0.80 0.202 1.5
## 
##                        PA3  PA1  PA4  PA5  PA6  PA2  PA7  PA8
## SS loadings           2.38 2.35 2.17 1.94 1.83 1.79 1.70 1.22
## Proportion Var        0.12 0.12 0.11 0.10 0.09 0.09 0.08 0.06
## Cumulative Var        0.12 0.24 0.34 0.44 0.53 0.62 0.71 0.77
## Proportion Explained  0.16 0.15 0.14 0.13 0.12 0.12 0.11 0.08
## Cumulative Proportion 0.16 0.31 0.45 0.57 0.69 0.81 0.92 1.00
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.43 with Chi Square =  4636.93
## df of  the model are 58  and the objective function was  0.36 
## 
## The root mean square of the residuals (RMSR) is  0.01 
## The df corrected root mean square of the residuals is  0.02 
## 
## The harmonic n.obs is  309 with the empirical chi square  15.22  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  106.28  with prob <  0.00011 
## 
## Tucker Lewis Index of factoring reliability =  0.964
## RMSEA index =  0.052  and the 90 % confidence intervals are  0.036 0.067
## BIC =  -226.25
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    PA3  PA1  PA4  PA5  PA6  PA2
## Correlation of (regression) scores with factors   0.93 0.95 0.92 0.93 0.99 0.98
## Multiple R square of scores with factors          0.87 0.91 0.85 0.87 0.98 0.96
## Minimum correlation of possible factor scores     0.75 0.82 0.70 0.73 0.96 0.92
##                                                    PA7  PA8
## Correlation of (regression) scores with factors   0.97 0.82
## Multiple R square of scores with factors          0.95 0.67
## Minimum correlation of possible factor scores     0.89 0.34

We can see that for 6 Factors Analysis, we obtain a cumulative proportion variance of 0.73. In total, the extracted factors explain 73% of the variance.

For 8 Factors Analysis, we obtain a cumulative proportion variance of 0.77. In total, the extracted factors explain 77% of the variance.

BIC is lower with 8 factors than 6 factors, therefore may allow more generalization in future sample.

We should also check the root mean square error of approximation (RMSEA). An better value should be closer to 0. In 6 Factors Analysis we have 0.067 and in 8 Factors Analysis we have 0.052 (closer to 0).

Finally, we must check the Tucker-Lewis Index (TLI). An acceptable value must be over 0.9. In 6 Factors Analysis we have 0.946 and in 8 Factors Analysis we have 0.964.

Therefore 8 Factors Analysis is overall better, with better BIC, RMSR and TLI and also explain more the total variance with 77%.

Choosing the Optimal Number of Factors in Exploratory Factor Analysis: A Model Selection Perspective

Labeling 8 Factors

Here we label the 8 factors according to the questions asked to receive the images which load on to the factor

colnames(fa_result8$loadings) <- c("Shopping Experience", "Store Decoration","Luxury Brands","French Culture","Product Assortment","Gourmet Food","Trendiness","Professionalism")

Shopping_Experience <- c("Im20","Im21","Im22")
Store_Decoration <- c("Im3","Im4","Im5")
Luxury_Brands <- c("Im11","Im12","Im13")
French_Culture <- c("Im6","Im7","Im9")
Product_Assortment <- c("Im1","Im2")
Gourmet_Food <- c("Im10","Im14")
Trendiness <- c("Im17","Im18")
Professionalism <- c("Im16","Im19")

And after renaming the factors we can now plot the loadings of the factors.

fa.diagram(fa_result8, sort = TRUE, adj = 1, rsize = 4, e.size = 0.061, main = "Conclusion of Factors Analysis - with 8 labeled factors", digits = 2, l.cex = 1)

Internal consistency reliability

Our next step is to assess the internal consistency reliability of the factors that were identified through the EFA. To accomplish this, we will use Cronbach’s alpha. We will evaluate the reliability of each factor individually by incorporating only the chosen items for that particular factor.

We need to look at:

1. Cronbach’s alpha (raw_alpha) which indicates the internal consistency reliability as well as 2. Corrected item-total correlation (average_r)

The interpretation is detailed as follows (DeVellis, 2012, pp. 95–96):

EFA and Cronbach’s alpha

Shopping Experience

alpha.pa1 <- psych::alpha(df_1[Shopping_Experience])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r     S/N        ase     mean       sd
##  0.8947029 0.8951786 0.8514649 0.7400356 8.54004 0.01029882 4.677454 1.341113
##   median_r
##  0.7296095

raw_alpha is over 0.7 and average items correlation is above 0.5

Store Decoration

alpha.pa1 <- psych::alpha(df_1[Store_Decoration])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9080535 0.9080107 0.8910999 0.7669149 9.870833 0.009512795 4.909385 1.251142
##  median_r
##  0.713708

raw_alpha is over 0.7 and average items correlation is above 0.5

Luxury Brands

alpha.pa1 <- psych::alpha(df_1[Luxury_Brands])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.8362383 0.8369592 0.7932492  0.631152 5.133432 0.01631641 5.549083 1.033799
##   median_r
##  0.5951255

raw_alpha is over 0.7 and average items correlation is above 0.5

French Culture

alpha.pa1 <- psych::alpha(df_1[French_Culture])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.7974734 0.8043581 0.7661015 0.5781409 4.111379 0.02082013 5.532902 1.061296
##   median_r
##  0.4812433

raw_alpha is over 0.7 and average items correlation is above 0.5

Product Assortment

alpha.pa1 <- psych::alpha(df_1[Product_Assortment])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean      sd
##  0.9370045 0.9377564 0.8828073 0.8828073 15.06591 0.007118763 4.847896 1.27965
##   median_r
##  0.8828073

raw_alpha is over 0.7 and average items correlation is above 0.5

Gourmet Food

alpha.pa1 <- psych::alpha(df_1[Gourmet_Food])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r     S/N         ase     mean        sd
##  0.9327078 0.9327084 0.8739021 0.8739021 13.8607 0.007656198 6.106796 0.8498963
##   median_r
##  0.8739021

raw_alpha is over 0.7 and average items correlation is above 0.5

Trendiness

alpha.pa1 <- psych::alpha(df_1[Trendiness])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9155341 0.9175086 0.8475898 0.8475898 11.12248 0.009476765 4.737864 1.287763
##   median_r
##  0.8475898

raw_alpha is over 0.7 and average items correlation is above 0.5

Professionalism

alpha.pa1 <- psych::alpha(df_1[Professionalism])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.8027054 0.8029503 0.6707744 0.6707744 4.074861 0.02242662 5.082524 1.119696
##   median_r
##  0.6707744

raw_alpha is over 0.7 and average items correlation is above 0.5

Our assessment suggests that the factors extracted are reliable, and therefore it is advisable to retain all the items related to these factors.

Dimensions by which Galeries Layfayette is perceived?

Here we show the dimensions by which galeries layfayette is perceived according to EFA.

fa.diagram(fa_result8, sort = TRUE, adj = 1, rsize = 4, e.size = 0.061, main = "Galeries Lafayette - Perception Dimensions", digits = 2, l.cex = 1)

Dimensions Definitions:

Product Assortment: This group pertains to the variety and range of products offered by the store.

Store Decoration: This group pertains to the aesthetic elements of the store’s interior and exterior, such as the artistic and creative decoration of the sales area, and the appealing arrangement of shop windows.

French Culture: This group pertains to elements of French culture, such as French savoir-vivre, fashion.

Gourmet Food: This group pertains to high-quality offered by the store.

Luxury Brands: This group pertains to the presence of luxury and designer brands in the store.

Professionalism: This group pertains to elements of professionalism, such as the store’s professional appearance towards customers and professional organization.

Trendiness: This group pertains to the store’s ability to stay current and up-to-date with the latest trends in the market.

Shopping Experience: This group pertains to the overall shopping experience, including elements such as relaxing shopping, a great place to stroll, and an intimate shop atmosphere.

Confirmatory Factor Analysis

In this part we will do a confirmatory factor analysis which is based on the results of EFA from before.

From Assistant For confirmatory factor analysis (CFA) and structural equation modeling (SEM), please use the raw data (which includes the missing values) to perform CFA and SEM, and use maximum likelihood (ML) to handle the missing data.

df <- read.csv2('Case Study III_Structural Equation Modeling.csv', na.strings = '999', sep = ',')

Summary of CFA Model with 8 Factors

Here we fit the CFA model and look at its summary.

model_CFA <-"
Shopping_Experience =~ Im20+Im22+Im21
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13+Im11
French_Culture =~ Im6+Im7+Im9
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 102 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        88
## 
##   Number of observations                           553
##   Number of missing patterns                        82
## 
## Model Test User Model:
##                                                       
##   Test statistic                               383.534
##   Degrees of freedom                               142
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7789.413
##   Degrees of freedom                               190
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.968
##   Tucker-Lewis Index (TLI)                       0.957
##                                                       
##   Robust Comparative Fit Index (CFI)             0.968
##   Robust Tucker-Lewis Index (TLI)                0.957
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -13802.030
##   Loglikelihood unrestricted model (H1)     -13610.263
##                                                       
##   Akaike (AIC)                               27780.060
##   Bayesian (BIC)                             28159.811
##   Sample-size adjusted Bayesian (SABIC)      27880.460
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.055
##   90 Percent confidence interval - lower         0.049
##   90 Percent confidence interval - upper         0.062
##   P-value H_0: RMSEA <= 0.050                    0.087
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.057
##   90 Percent confidence interval - lower         0.050
##   90 Percent confidence interval - upper         0.064
##   P-value H_0: Robust RMSEA <= 0.050             0.055
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.052
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.264    0.845
##     Im22                    1.060    0.047   22.590    0.000    1.341    0.877
##     Im21                    0.849    0.041   20.818    0.000    1.073    0.783
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.236    0.937
##     Im4                     1.056    0.025   42.717    0.000    1.305    0.969
##     Im5                     0.818    0.034   23.813    0.000    1.011    0.760
##   Luxury_Brands =~                                                            
##     Im12                    1.000                               0.991    0.872
##     Im13                    1.038    0.050   20.653    0.000    1.029    0.855
##     Im11                    0.709    0.047   15.048    0.000    0.703    0.615
##   French_Culture =~                                                           
##     Im6                     1.000                               1.002    0.835
##     Im7                     1.107    0.050   22.219    0.000    1.109    0.919
##     Im9                     0.789    0.057   13.916    0.000    0.790    0.585
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.305    0.980
##     Im2                     0.885    0.033   27.039    0.000    1.155    0.899
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.812    0.924
##     Im14                    1.014    0.035   28.587    0.000    0.823    0.952
##   Trendiness =~                                                               
##     Im17                    1.000                               1.204    0.968
##     Im18                    0.995    0.041   24.250    0.000    1.197    0.857
##   Professionalism =~                                                          
##     Im16                    1.000                               0.922    0.766
##     Im19                    1.045    0.061   17.188    0.000    0.963    0.856
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.729    0.082    8.911    0.000    0.467    0.467
##     Luxury_Brands           0.524    0.068    7.693    0.000    0.418    0.418
##     French_Culture          0.446    0.066    6.749    0.000    0.352    0.352
##     Prdct_Assrtmnt          0.739    0.085    8.728    0.000    0.448    0.448
##     Gourmet_Food            0.303    0.051    5.951    0.000    0.295    0.295
##     Trendiness              0.786    0.081    9.715    0.000    0.517    0.517
##     Professionalsm          0.557    0.069    8.091    0.000    0.478    0.478
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.577    0.064    8.957    0.000    0.471    0.471
##     French_Culture          0.449    0.063    7.099    0.000    0.363    0.363
##     Prdct_Assrtmnt          0.711    0.079    9.032    0.000    0.440    0.440
##     Gourmet_Food            0.418    0.050    8.401    0.000    0.417    0.417
##     Trendiness              0.770    0.076   10.141    0.000    0.517    0.517
##     Professionalsm          0.744    0.071   10.469    0.000    0.653    0.653
##   Luxury_Brands ~~                                                            
##     French_Culture          0.337    0.052    6.460    0.000    0.340    0.340
##     Prdct_Assrtmnt          0.618    0.068    9.135    0.000    0.478    0.478
##     Gourmet_Food            0.364    0.043    8.499    0.000    0.452    0.452
##     Trendiness              0.676    0.065   10.423    0.000    0.566    0.566
##     Professionalsm          0.483    0.055    8.826    0.000    0.529    0.529
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.321    0.063    5.121    0.000    0.246    0.246
##     Gourmet_Food            0.490    0.047   10.536    0.000    0.603    0.603
##     Trendiness              0.439    0.062    7.087    0.000    0.364    0.364
##     Professionalsm          0.360    0.052    6.937    0.000    0.391    0.391
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.328    0.050    6.581    0.000    0.309    0.309
##     Trendiness              0.817    0.079   10.362    0.000    0.519    0.519
##     Professionalsm          0.718    0.072    9.961    0.000    0.597    0.597
##   Gourmet_Food ~~                                                             
##     Trendiness              0.318    0.047    6.804    0.000    0.325    0.325
##     Professionalsm          0.373    0.043    8.600    0.000    0.498    0.498
##   Trendiness ~~                                                               
##     Professionalsm          0.667    0.066   10.043    0.000    0.601    0.601
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.178    0.000    4.672    3.123
##    .Im22              4.279    0.065   65.401    0.000    4.279    2.799
##    .Im21              5.139    0.058   87.970    0.000    5.139    3.751
##    .Im3               4.995    0.056   88.565    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.988    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.848    0.000    5.035    3.787
##    .Im12              5.666    0.049  116.093    0.000    5.666    4.983
##    .Im13              5.448    0.052  105.619    0.000    5.448    4.524
##    .Im11              5.653    0.049  115.273    0.000    5.653    4.943
##    .Im6               5.826    0.051  113.774    0.000    5.826    4.856
##    .Im7               5.751    0.052  111.068    0.000    5.751    4.766
##    .Im9               5.075    0.058   87.408    0.000    5.075    3.756
##    .Im1               4.790    0.057   84.203    0.000    4.790    3.597
##    .Im2               4.857    0.055   88.356    0.000    4.857    3.779
##    .Im10              6.100    0.037  162.799    0.000    6.100    6.937
##    .Im14              6.138    0.037  165.865    0.000    6.138    7.093
##    .Im17              5.025    0.053   94.529    0.000    5.025    4.041
##    .Im18              4.595    0.060   76.455    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.150    0.000    5.135    4.269
##    .Im19              5.145    0.048  106.954    0.000    5.145    4.574
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.639    0.061   10.445    0.000    0.639    0.285
##    .Im22              0.540    0.063    8.515    0.000    0.540    0.231
##    .Im21              0.726    0.057   12.672    0.000    0.726    0.387
##    .Im3               0.213    0.024    8.760    0.000    0.213    0.123
##    .Im4               0.109    0.024    4.524    0.000    0.109    0.060
##    .Im5               0.747    0.049   15.217    0.000    0.747    0.422
##    .Im12              0.310    0.040    7.833    0.000    0.310    0.239
##    .Im13              0.390    0.045    8.771    0.000    0.390    0.269
##    .Im11              0.814    0.055   14.803    0.000    0.814    0.622
##    .Im6               0.436    0.042   10.443    0.000    0.436    0.303
##    .Im7               0.227    0.042    5.343    0.000    0.227    0.156
##    .Im9               1.201    0.080   15.032    0.000    1.201    0.658
##    .Im1               0.070    0.050    1.383    0.167    0.070    0.039
##    .Im2               0.317    0.044    7.242    0.000    0.317    0.192
##    .Im10              0.113    0.019    5.935    0.000    0.113    0.146
##    .Im14              0.071    0.019    3.764    0.000    0.071    0.094
##    .Im17              0.096    0.045    2.153    0.031    0.096    0.062
##    .Im18              0.520    0.054    9.566    0.000    0.520    0.266
##    .Im16              0.598    0.052   11.498    0.000    0.598    0.413
##    .Im19              0.338    0.045    7.481    0.000    0.338    0.267
##     Shoppng_Exprnc    1.599    0.138   11.620    0.000    1.000    1.000
##     Store_Decoratn    1.527    0.107   14.326    0.000    1.000    1.000
##     Luxury_Brands     0.983    0.084   11.680    0.000    1.000    1.000
##     French_Culture    1.003    0.089   11.297    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.704    0.118   14.391    0.000    1.000    1.000
##     Gourmet_Food      0.660    0.049   13.357    0.000    1.000    1.000
##     Trendiness        1.450    0.104   13.998    0.000    1.000    1.000
##     Professionalsm    0.849    0.088    9.644    0.000    1.000    1.000

Modification indices of CFA Model with 8 Factor

modificationindices(fit_CFA) %>% filter(mi>10)
##                    lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1  Shopping_Experience =~ Im12 11.029 -0.115  -0.145   -0.128   -0.128
## 2  Shopping_Experience =~  Im9 20.552  0.204   0.258    0.191    0.191
## 3     Store_Decoration =~  Im9 19.199  0.194   0.240    0.178    0.178
## 4        Luxury_Brands =~  Im6 11.249 -0.140  -0.139   -0.116   -0.116
## 5        Luxury_Brands =~  Im9 74.107  0.496   0.492    0.364    0.364
## 6   Product_Assortment =~ Im20 14.739 -0.151  -0.197   -0.131   -0.131
## 7   Product_Assortment =~ Im12 10.703 -0.111  -0.145   -0.128   -0.128
## 8   Product_Assortment =~ Im13 14.024  0.133   0.174    0.144    0.144
## 9   Product_Assortment =~  Im9 22.723  0.190   0.248    0.183    0.183
## 10        Gourmet_Food =~ Im11 12.787  0.215   0.175    0.153    0.153
## 11        Gourmet_Food =~  Im6 11.064 -0.237  -0.193   -0.161   -0.161
## 12          Trendiness =~ Im12 17.366 -0.180  -0.217   -0.190   -0.190
## 13          Trendiness =~ Im13 23.945  0.220   0.265    0.220    0.220
## 14          Trendiness =~  Im7 17.707 -0.148  -0.178   -0.148   -0.148
## 15          Trendiness =~  Im9 58.323  0.350   0.422    0.312    0.312
## 16     Professionalism =~  Im9 28.665  0.349   0.322    0.238    0.238
## 17                Im20 ~~ Im21 11.693  0.231   0.231    0.339    0.339
## 18                Im22 ~~ Im21 15.187 -0.286  -0.286   -0.457   -0.457
## 19                Im12 ~~ Im11 13.264  0.145   0.145    0.288    0.288
## 20                Im13 ~~ Im11 21.231 -0.190  -0.190   -0.338   -0.338
## 21                Im13 ~~  Im1 10.737  0.068   0.068    0.410    0.410
## 22                 Im6 ~~  Im7 26.792  0.475   0.475    1.511    1.511

Summary of CFA Model with 8 Factors - Removing Im9

model_CFA <-"
Shopping_Experience =~ Im20+Im22+Im21
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13+Im11
French_Culture =~ Im6+Im7
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

# removed: Im9

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 106 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        85
## 
##   Number of observations                           553
##   Number of missing patterns                        79
## 
## Model Test User Model:
##                                                       
##   Test statistic                               259.047
##   Degrees of freedom                               124
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7474.765
##   Degrees of freedom                               171
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.982
##   Tucker-Lewis Index (TLI)                       0.975
##                                                       
##   Robust Comparative Fit Index (CFI)             0.981
##   Robust Tucker-Lewis Index (TLI)                0.974
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -12973.111
##   Loglikelihood unrestricted model (H1)     -12843.588
##                                                       
##   Akaike (AIC)                               26116.223
##   Bayesian (BIC)                             26483.028
##   Sample-size adjusted Bayesian (SABIC)      26213.200
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.044
##   90 Percent confidence interval - lower         0.037
##   90 Percent confidence interval - upper         0.052
##   P-value H_0: RMSEA <= 0.050                    0.886
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.045
##   90 Percent confidence interval - lower         0.038
##   90 Percent confidence interval - upper         0.053
##   P-value H_0: Robust RMSEA <= 0.050             0.825
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.029
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.265    0.845
##     Im22                    1.060    0.047   22.606    0.000    1.340    0.877
##     Im21                    0.849    0.041   20.824    0.000    1.074    0.783
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.236    0.937
##     Im4                     1.056    0.025   42.716    0.000    1.305    0.969
##     Im5                     0.818    0.034   23.815    0.000    1.011    0.760
##   Luxury_Brands =~                                                            
##     Im12                    1.000                               0.991    0.872
##     Im13                    1.039    0.050   20.658    0.000    1.030    0.855
##     Im11                    0.709    0.047   15.046    0.000    0.703    0.615
##   French_Culture =~                                                           
##     Im6                     1.000                               0.975    0.813
##     Im7                     1.184    0.071   16.770    0.000    1.155    0.955
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.305    0.980
##     Im2                     0.885    0.033   27.043    0.000    1.155    0.899
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.812    0.923
##     Im14                    1.015    0.036   28.479    0.000    0.824    0.952
##   Trendiness =~                                                               
##     Im17                    1.000                               1.204    0.969
##     Im18                    0.994    0.041   24.143    0.000    1.197    0.856
##   Professionalism =~                                                          
##     Im16                    1.000                               0.921    0.766
##     Im19                    1.046    0.061   17.170    0.000    0.963    0.856
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.730    0.082    8.912    0.000    0.467    0.467
##     Luxury_Brands           0.524    0.068    7.696    0.000    0.418    0.418
##     French_Culture          0.410    0.065    6.352    0.000    0.333    0.333
##     Prdct_Assrtmnt          0.739    0.085    8.728    0.000    0.448    0.448
##     Gourmet_Food            0.303    0.051    5.948    0.000    0.295    0.295
##     Trendiness              0.787    0.081    9.715    0.000    0.516    0.516
##     Professionalsm          0.557    0.069    8.089    0.000    0.478    0.478
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.577    0.064    8.958    0.000    0.471    0.471
##     French_Culture          0.402    0.063    6.350    0.000    0.334    0.334
##     Prdct_Assrtmnt          0.711    0.079    9.032    0.000    0.441    0.441
##     Gourmet_Food            0.418    0.050    8.393    0.000    0.416    0.416
##     Trendiness              0.770    0.076   10.140    0.000    0.517    0.517
##     Professionalsm          0.743    0.071   10.465    0.000    0.653    0.653
##   Luxury_Brands ~~                                                            
##     French_Culture          0.295    0.050    5.922    0.000    0.306    0.306
##     Prdct_Assrtmnt          0.618    0.068    9.135    0.000    0.478    0.478
##     Gourmet_Food            0.364    0.043    8.495    0.000    0.452    0.452
##     Trendiness              0.676    0.065   10.425    0.000    0.566    0.566
##     Professionalsm          0.483    0.055    8.825    0.000    0.529    0.529
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.286    0.060    4.735    0.000    0.225    0.225
##     Gourmet_Food            0.463    0.047    9.829    0.000    0.585    0.585
##     Trendiness              0.378    0.061    6.175    0.000    0.322    0.322
##     Professionalsm          0.328    0.051    6.438    0.000    0.366    0.366
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.328    0.050    6.584    0.000    0.309    0.309
##     Trendiness              0.817    0.079   10.362    0.000    0.519    0.519
##     Professionalsm          0.717    0.072    9.956    0.000    0.597    0.597
##   Gourmet_Food ~~                                                             
##     Trendiness              0.318    0.047    6.801    0.000    0.325    0.325
##     Professionalsm          0.372    0.043    8.589    0.000    0.498    0.498
##   Trendiness ~~                                                               
##     Professionalsm          0.667    0.066   10.040    0.000    0.601    0.601
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.177    0.000    4.672    3.123
##    .Im22              4.279    0.065   65.401    0.000    4.279    2.799
##    .Im21              5.139    0.058   87.970    0.000    5.139    3.751
##    .Im3               4.995    0.056   88.560    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.983    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.844    0.000    5.035    3.787
##    .Im12              5.666    0.049  116.089    0.000    5.666    4.983
##    .Im13              5.448    0.052  105.615    0.000    5.448    4.524
##    .Im11              5.653    0.049  115.271    0.000    5.653    4.943
##    .Im6               5.827    0.051  113.784    0.000    5.827    4.858
##    .Im7               5.753    0.052  110.826    0.000    5.753    4.756
##    .Im1               4.790    0.057   84.202    0.000    4.790    3.597
##    .Im2               4.857    0.055   88.354    0.000    4.857    3.779
##    .Im10              6.100    0.037  162.789    0.000    6.100    6.937
##    .Im14              6.138    0.037  165.861    0.000    6.138    7.093
##    .Im17              5.025    0.053   94.519    0.000    5.025    4.041
##    .Im18              4.595    0.060   76.447    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.147    0.000    5.135    4.269
##    .Im19              5.145    0.048  106.948    0.000    5.145    4.574
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.638    0.061   10.451    0.000    0.638    0.285
##    .Im22              0.541    0.063    8.540    0.000    0.541    0.231
##    .Im21              0.725    0.057   12.672    0.000    0.725    0.386
##    .Im3               0.213    0.024    8.755    0.000    0.213    0.122
##    .Im4               0.109    0.024    4.532    0.000    0.109    0.060
##    .Im5               0.747    0.049   15.217    0.000    0.747    0.422
##    .Im12              0.310    0.040    7.845    0.000    0.310    0.240
##    .Im13              0.390    0.045    8.765    0.000    0.390    0.269
##    .Im11              0.814    0.055   14.802    0.000    0.814    0.622
##    .Im6               0.487    0.056    8.677    0.000    0.487    0.339
##    .Im7               0.128    0.067    1.930    0.054    0.128    0.088
##    .Im1               0.070    0.050    1.394    0.163    0.070    0.040
##    .Im2               0.317    0.044    7.233    0.000    0.317    0.192
##    .Im10              0.114    0.019    5.961    0.000    0.114    0.148
##    .Im14              0.070    0.019    3.680    0.000    0.070    0.093
##    .Im17              0.095    0.045    2.112    0.035    0.095    0.062
##    .Im18              0.521    0.055    9.540    0.000    0.521    0.267
##    .Im16              0.599    0.052   11.498    0.000    0.599    0.414
##    .Im19              0.338    0.045    7.457    0.000    0.338    0.267
##     Shoppng_Exprnc    1.599    0.138   11.623    0.000    1.000    1.000
##     Store_Decoratn    1.528    0.107   14.326    0.000    1.000    1.000
##     Luxury_Brands     0.983    0.084   11.678    0.000    1.000    1.000
##     French_Culture    0.952    0.095   10.058    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.704    0.118   14.388    0.000    1.000    1.000
##     Gourmet_Food      0.659    0.049   13.328    0.000    1.000    1.000
##     Trendiness        1.451    0.104   13.988    0.000    1.000    1.000
##     Professionalsm    0.849    0.088    9.638    0.000    1.000    1.000

Modification indices of CFA Model with 8 Factors - Removing Im9

modificationindices(fit_CFA) %>% filter(mi>10)
##                    lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1  Shopping_Experience =~ Im12 10.952 -0.115  -0.145   -0.127   -0.127
## 2   Product_Assortment =~ Im20 14.777 -0.151  -0.197   -0.132   -0.132
## 3   Product_Assortment =~ Im12 10.663 -0.111  -0.145   -0.127   -0.127
## 4   Product_Assortment =~ Im13 13.970  0.133   0.174    0.144    0.144
## 5         Gourmet_Food =~ Im11 12.742  0.215   0.174    0.152    0.152
## 6           Trendiness =~ Im12 17.245 -0.179  -0.216   -0.190   -0.190
## 7           Trendiness =~ Im13 23.832  0.220   0.265    0.220    0.220
## 8                 Im20 ~~ Im21 11.455  0.228   0.228    0.335    0.335
## 9                 Im22 ~~ Im21 15.139 -0.285  -0.285   -0.455   -0.455
## 10                Im12 ~~ Im11 13.307  0.145   0.145    0.288    0.288
## 11                Im13 ~~ Im11 21.323 -0.191  -0.191   -0.338   -0.338
## 12                Im13 ~~  Im1 10.707  0.068   0.068    0.409    0.409

Summary of CFA Model with 8 Factors - Removing Im9+Im11

model_CFA <-"
Shopping_Experience =~ Im20+Im22+Im21
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13
French_Culture =~ Im6+Im7
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

# removed: Im9, Im11

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 104 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        82
## 
##   Number of observations                           553
##   Number of missing patterns                        75
## 
## Model Test User Model:
##                                                       
##   Test statistic                               203.508
##   Degrees of freedom                               107
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7217.692
##   Degrees of freedom                               153
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.986
##   Tucker-Lewis Index (TLI)                       0.980
##                                                       
##   Robust Comparative Fit Index (CFI)             0.986
##   Robust Tucker-Lewis Index (TLI)                0.980
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -12234.854
##   Loglikelihood unrestricted model (H1)     -12133.100
##                                                       
##   Akaike (AIC)                               24633.709
##   Bayesian (BIC)                             24987.568
##   Sample-size adjusted Bayesian (SABIC)      24727.264
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.040
##   90 Percent confidence interval - lower         0.032
##   90 Percent confidence interval - upper         0.049
##   P-value H_0: RMSEA <= 0.050                    0.971
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.041
##   90 Percent confidence interval - lower         0.033
##   90 Percent confidence interval - upper         0.050
##   P-value H_0: Robust RMSEA <= 0.050             0.946
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.024
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.264    0.845
##     Im22                    1.061    0.047   22.572    0.000    1.341    0.877
##     Im21                    0.849    0.041   20.811    0.000    1.073    0.783
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.236    0.937
##     Im4                     1.056    0.025   42.718    0.000    1.305    0.969
##     Im5                     0.818    0.034   23.812    0.000    1.011    0.760
##   Luxury_Brands =~                                                            
##     Im12                    1.000                               0.925    0.814
##     Im13                    1.197    0.068   17.500    0.000    1.108    0.919
##   French_Culture =~                                                           
##     Im6                     1.000                               0.975    0.813
##     Im7                     1.185    0.070   16.849    0.000    1.155    0.955
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.309    0.983
##     Im2                     0.880    0.033   26.989    0.000    1.152    0.896
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.811    0.922
##     Im14                    1.018    0.036   28.129    0.000    0.825    0.954
##   Trendiness =~                                                               
##     Im17                    1.000                               1.208    0.971
##     Im18                    0.989    0.041   24.254    0.000    1.194    0.854
##   Professionalism =~                                                          
##     Im16                    1.000                               0.921    0.766
##     Im19                    1.046    0.061   17.135    0.000    0.964    0.857
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.729    0.082    8.912    0.000    0.467    0.467
##     Luxury_Brands           0.475    0.065    7.311    0.000    0.407    0.407
##     French_Culture          0.410    0.065    6.356    0.000    0.333    0.333
##     Prdct_Assrtmnt          0.741    0.085    8.745    0.000    0.448    0.448
##     Gourmet_Food            0.302    0.051    5.941    0.000    0.295    0.295
##     Trendiness              0.786    0.081    9.708    0.000    0.515    0.515
##     Professionalsm          0.556    0.069    8.084    0.000    0.478    0.478
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.529    0.063    8.358    0.000    0.463    0.463
##     French_Culture          0.402    0.063    6.355    0.000    0.334    0.334
##     Prdct_Assrtmnt          0.711    0.079    9.035    0.000    0.440    0.440
##     Gourmet_Food            0.417    0.050    8.384    0.000    0.416    0.416
##     Trendiness              0.770    0.076   10.133    0.000    0.516    0.516
##     Professionalsm          0.743    0.071   10.458    0.000    0.653    0.653
##   Luxury_Brands ~~                                                            
##     French_Culture          0.256    0.048    5.378    0.000    0.284    0.284
##     Prdct_Assrtmnt          0.592    0.066    8.971    0.000    0.489    0.489
##     Gourmet_Food            0.310    0.042    7.391    0.000    0.413    0.413
##     Trendiness              0.646    0.064   10.050    0.000    0.579    0.579
##     Professionalsm          0.441    0.054    8.215    0.000    0.517    0.517
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.286    0.061    4.723    0.000    0.224    0.224
##     Gourmet_Food            0.463    0.047    9.822    0.000    0.585    0.585
##     Trendiness              0.378    0.061    6.179    0.000    0.321    0.321
##     Professionalsm          0.328    0.051    6.441    0.000    0.366    0.366
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.327    0.050    6.578    0.000    0.309    0.309
##     Trendiness              0.817    0.079   10.366    0.000    0.517    0.517
##     Professionalsm          0.717    0.072    9.945    0.000    0.595    0.595
##   Gourmet_Food ~~                                                             
##     Trendiness              0.318    0.047    6.810    0.000    0.325    0.325
##     Professionalsm          0.371    0.043    8.565    0.000    0.497    0.497
##   Trendiness ~~                                                               
##     Professionalsm          0.668    0.066   10.050    0.000    0.600    0.600
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.175    0.000    4.672    3.123
##    .Im22              4.279    0.065   65.401    0.000    4.279    2.799
##    .Im21              5.139    0.058   87.969    0.000    5.139    3.751
##    .Im3               4.995    0.056   88.561    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.984    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.844    0.000    5.035    3.787
##    .Im12              5.665    0.049  116.049    0.000    5.665    4.986
##    .Im13              5.448    0.052  105.546    0.000    5.448    4.521
##    .Im6               5.827    0.051  113.785    0.000    5.827    4.858
##    .Im7               5.753    0.052  110.824    0.000    5.753    4.756
##    .Im1               4.791    0.057   84.201    0.000    4.791    3.597
##    .Im2               4.857    0.055   88.354    0.000    4.857    3.779
##    .Im10              6.100    0.037  162.776    0.000    6.100    6.936
##    .Im14              6.138    0.037  165.836    0.000    6.138    7.092
##    .Im17              5.025    0.053   94.523    0.000    5.025    4.041
##    .Im18              4.595    0.060   76.454    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.142    0.000    5.135    4.269
##    .Im19              5.145    0.048  106.947    0.000    5.145    4.574
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.640    0.061   10.459    0.000    0.640    0.286
##    .Im22              0.539    0.063    8.489    0.000    0.539    0.231
##    .Im21              0.726    0.057   12.661    0.000    0.726    0.386
##    .Im3               0.213    0.024    8.752    0.000    0.213    0.122
##    .Im4               0.109    0.024    4.527    0.000    0.109    0.060
##    .Im5               0.747    0.049   15.217    0.000    0.747    0.422
##    .Im12              0.435    0.047    9.159    0.000    0.435    0.337
##    .Im13              0.226    0.058    3.892    0.000    0.226    0.155
##    .Im6               0.488    0.056    8.724    0.000    0.488    0.339
##    .Im7               0.128    0.066    1.939    0.053    0.128    0.088
##    .Im1               0.060    0.051    1.183    0.237    0.060    0.034
##    .Im2               0.325    0.044    7.411    0.000    0.325    0.197
##    .Im10              0.116    0.020    5.959    0.000    0.116    0.150
##    .Im14              0.068    0.019    3.503    0.000    0.068    0.090
##    .Im17              0.088    0.045    1.957    0.050    0.088    0.057
##    .Im18              0.528    0.054    9.725    0.000    0.528    0.270
##    .Im16              0.599    0.052   11.487    0.000    0.599    0.414
##    .Im19              0.337    0.045    7.424    0.000    0.337    0.266
##     Shoppng_Exprnc    1.597    0.138   11.609    0.000    1.000    1.000
##     Store_Decoratn    1.527    0.107   14.325    0.000    1.000    1.000
##     Luxury_Brands     0.856    0.084   10.186    0.000    1.000    1.000
##     French_Culture    0.951    0.094   10.075    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.714    0.119   14.453    0.000    1.000    1.000
##     Gourmet_Food      0.657    0.050   13.263    0.000    1.000    1.000
##     Trendiness        1.459    0.104   14.068    0.000    1.000    1.000
##     Professionalsm    0.848    0.088    9.629    0.000    1.000    1.000

Modification indices of CFA Model with 8 Factors - Removing Im9+Im11

modificationindices(fit_CFA) %>% filter(mi>10)
##                  lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1 Product_Assortment =~ Im20 14.627 -0.149  -0.195   -0.131   -0.131
## 2       Gourmet_Food =~ Im12 11.344  0.186   0.151    0.133    0.133
## 3       Gourmet_Food =~ Im13 11.344 -0.222  -0.180   -0.150   -0.150
## 4               Im20 ~~ Im21 11.943  0.233   0.233    0.342    0.342
## 5               Im22 ~~ Im21 15.754 -0.292  -0.292   -0.466   -0.466

Summary of CFA Model with 8 Factors - Removing Im9+Im11+Im21

model_CFA <-"
Shopping_Experience =~ Im20+Im22
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13
French_Culture =~ Im6+Im7
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

# removed Im9, Im11, Im21

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 105 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        79
## 
##   Number of observations                           553
##   Number of missing patterns                        72
## 
## Model Test User Model:
##                                                       
##   Test statistic                               171.890
##   Degrees of freedom                                91
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              6788.564
##   Degrees of freedom                               136
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.988
##   Tucker-Lewis Index (TLI)                       0.982
##                                                       
##   Robust Comparative Fit Index (CFI)             0.988
##   Robust Tucker-Lewis Index (TLI)                0.982
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -11483.125
##   Loglikelihood unrestricted model (H1)     -11397.180
##                                                       
##   Akaike (AIC)                               23124.249
##   Bayesian (BIC)                             23465.163
##   Sample-size adjusted Bayesian (SABIC)      23214.382
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.040
##   90 Percent confidence interval - lower         0.031
##   90 Percent confidence interval - upper         0.049
##   P-value H_0: RMSEA <= 0.050                    0.964
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.041
##   90 Percent confidence interval - lower         0.032
##   90 Percent confidence interval - upper         0.051
##   P-value H_0: Robust RMSEA <= 0.050             0.937
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.025
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.136    0.759
##     Im22                    1.319    0.092   14.335    0.000    1.499    0.979
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.237    0.937
##     Im4                     1.055    0.025   42.733    0.000    1.304    0.969
##     Im5                     0.817    0.034   23.822    0.000    1.011    0.760
##   Luxury_Brands =~                                                            
##     Im12                    1.000                               0.926    0.815
##     Im13                    1.195    0.068   17.469    0.000    1.107    0.918
##   French_Culture =~                                                           
##     Im6                     1.000                               0.976    0.814
##     Im7                     1.182    0.069   17.097    0.000    1.154    0.954
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.312    0.986
##     Im2                     0.875    0.032   26.977    0.000    1.149    0.894
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.811    0.922
##     Im14                    1.018    0.036   28.124    0.000    0.826    0.954
##   Trendiness =~                                                               
##     Im17                    1.000                               1.205    0.970
##     Im18                    0.992    0.041   24.481    0.000    1.195    0.855
##   Professionalism =~                                                          
##     Im16                    1.000                               0.921    0.765
##     Im19                    1.047    0.061   17.109    0.000    0.964    0.857
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.633    0.082    7.726    0.000    0.451    0.451
##     Luxury_Brands           0.366    0.064    5.712    0.000    0.348    0.348
##     French_Culture          0.375    0.060    6.226    0.000    0.339    0.339
##     Prdct_Assrtmnt          0.640    0.082    7.797    0.000    0.429    0.429
##     Gourmet_Food            0.249    0.047    5.278    0.000    0.270    0.270
##     Trendiness              0.687    0.082    8.359    0.000    0.502    0.502
##     Professionalsm          0.462    0.068    6.807    0.000    0.441    0.441
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.530    0.063    8.364    0.000    0.463    0.463
##     French_Culture          0.403    0.063    6.380    0.000    0.334    0.334
##     Prdct_Assrtmnt          0.711    0.079    9.032    0.000    0.438    0.438
##     Gourmet_Food            0.418    0.050    8.386    0.000    0.416    0.416
##     Trendiness              0.770    0.076   10.140    0.000    0.517    0.517
##     Professionalsm          0.744    0.071   10.457    0.000    0.653    0.653
##   Luxury_Brands ~~                                                            
##     French_Culture          0.256    0.048    5.381    0.000    0.284    0.284
##     Prdct_Assrtmnt          0.593    0.066    8.993    0.000    0.488    0.488
##     Gourmet_Food            0.310    0.042    7.394    0.000    0.413    0.413
##     Trendiness              0.645    0.064   10.045    0.000    0.578    0.578
##     Professionalsm          0.441    0.054    8.216    0.000    0.517    0.517
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.285    0.061    4.710    0.000    0.223    0.223
##     Gourmet_Food            0.463    0.047    9.847    0.000    0.585    0.585
##     Trendiness              0.378    0.061    6.197    0.000    0.322    0.322
##     Professionalsm          0.329    0.051    6.455    0.000    0.366    0.366
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.327    0.050    6.560    0.000    0.307    0.307
##     Trendiness              0.816    0.079   10.354    0.000    0.516    0.516
##     Professionalsm          0.716    0.072    9.933    0.000    0.593    0.593
##   Gourmet_Food ~~                                                             
##     Trendiness              0.317    0.047    6.796    0.000    0.325    0.325
##     Professionalsm          0.371    0.043    8.562    0.000    0.497    0.497
##   Trendiness ~~                                                               
##     Professionalsm          0.667    0.066   10.039    0.000    0.601    0.601
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.670    0.064   73.048    0.000    4.670    3.119
##    .Im22              4.280    0.065   65.348    0.000    4.280    2.797
##    .Im3               4.995    0.056   88.565    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.989    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.842    0.000    5.035    3.787
##    .Im12              5.665    0.049  116.075    0.000    5.665    4.987
##    .Im13              5.448    0.052  105.567    0.000    5.448    4.522
##    .Im6               5.827    0.051  113.783    0.000    5.827    4.858
##    .Im7               5.753    0.052  110.866    0.000    5.753    4.758
##    .Im1               4.791    0.057   84.238    0.000    4.791    3.598
##    .Im2               4.857    0.055   88.364    0.000    4.857    3.779
##    .Im10              6.100    0.037  162.765    0.000    6.100    6.936
##    .Im14              6.138    0.037  165.818    0.000    6.138    7.091
##    .Im17              5.025    0.053   94.570    0.000    5.025    4.043
##    .Im18              4.595    0.060   76.477    0.000    4.595    3.288
##    .Im16              5.136    0.052   99.141    0.000    5.136    4.269
##    .Im19              5.145    0.048  106.952    0.000    5.145    4.574
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.952    0.096    9.941    0.000    0.952    0.425
##    .Im22              0.096    0.133    0.721    0.471    0.096    0.041
##    .Im3               0.211    0.024    8.679    0.000    0.211    0.121
##    .Im4               0.112    0.024    4.637    0.000    0.112    0.062
##    .Im5               0.746    0.049   15.214    0.000    0.746    0.422
##    .Im12              0.433    0.048    9.111    0.000    0.433    0.336
##    .Im13              0.227    0.058    3.915    0.000    0.227    0.157
##    .Im6               0.486    0.055    8.821    0.000    0.486    0.338
##    .Im7               0.130    0.065    2.008    0.045    0.130    0.089
##    .Im1               0.050    0.051    0.993    0.321    0.050    0.028
##    .Im2               0.332    0.044    7.577    0.000    0.332    0.201
##    .Im10              0.116    0.020    5.966    0.000    0.116    0.150
##    .Im14              0.067    0.019    3.496    0.000    0.067    0.090
##    .Im17              0.092    0.044    2.086    0.037    0.092    0.060
##    .Im18              0.524    0.054    9.744    0.000    0.524    0.268
##    .Im16              0.600    0.052   11.482    0.000    0.600    0.414
##    .Im19              0.336    0.045    7.394    0.000    0.336    0.266
##     Shoppng_Exprnc    1.290    0.144    8.955    0.000    1.000    1.000
##     Store_Decoratn    1.530    0.107   14.344    0.000    1.000    1.000
##     Luxury_Brands     0.857    0.084   10.189    0.000    1.000    1.000
##     French_Culture    0.953    0.094   10.133    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.722    0.119   14.527    0.000    1.000    1.000
##     Gourmet_Food      0.657    0.050   13.260    0.000    1.000    1.000
##     Trendiness        1.453    0.103   14.075    0.000    1.000    1.000
##     Professionalsm    0.848    0.088    9.619    0.000    1.000    1.000

Modification indices of CFA Model with 8 Factors - Removing Im9+Im11+Im21

modificationindices(fit_CFA) %>% filter(mi>10)
##             lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1 Luxury_Brands =~ Im20 20.767  0.296   0.274    0.183    0.183
## 2 Luxury_Brands =~ Im22 20.767 -0.390  -0.361   -0.236   -0.236
## 3  Gourmet_Food =~ Im12 11.239  0.185   0.150    0.132    0.132
## 4  Gourmet_Food =~ Im13 11.239 -0.222  -0.180   -0.149   -0.149

Summary of CFA Model with 7 Factors - Removing Im9+Im11 and Shopping_Experience Factor

model_CFA <-"
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13
French_Culture =~ Im6+Im7
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

# removed Im9, Im11, Im21, Shopping_Experience =~ Im20+Im22

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 95 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        66
## 
##   Number of observations                           553
##   Number of missing patterns                        64
## 
## Model Test User Model:
##                                                       
##   Test statistic                               126.805
##   Degrees of freedom                                69
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              6104.996
##   Degrees of freedom                               105
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.990
##   Tucker-Lewis Index (TLI)                       0.985
##                                                       
##   Robust Comparative Fit Index (CFI)             0.990
##   Robust Tucker-Lewis Index (TLI)                0.985
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -9820.987
##   Loglikelihood unrestricted model (H1)      -9757.585
##                                                       
##   Akaike (AIC)                               19773.975
##   Bayesian (BIC)                             20058.788
##   Sample-size adjusted Bayesian (SABIC)      19849.275
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.039
##   90 Percent confidence interval - lower         0.028
##   90 Percent confidence interval - upper         0.049
##   P-value H_0: RMSEA <= 0.050                    0.958
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.040
##   90 Percent confidence interval - lower         0.029
##   90 Percent confidence interval - upper         0.051
##   P-value H_0: Robust RMSEA <= 0.050             0.930
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.022
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                         Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Store_Decoration =~                                                        
##     Im3                    1.000                               1.235    0.936
##     Im4                    1.059    0.025   42.544    0.000    1.307    0.971
##     Im5                    0.818    0.034   23.770    0.000    1.010    0.759
##   Luxury_Brands =~                                                           
##     Im12                   1.000                               0.926    0.815
##     Im13                   1.195    0.068   17.480    0.000    1.107    0.918
##   French_Culture =~                                                          
##     Im6                    1.000                               0.983    0.820
##     Im7                    1.167    0.070   16.701    0.000    1.147    0.948
##   Product_Assortment =~                                                      
##     Im1                    1.000                               1.302    0.978
##     Im2                    0.889    0.033   26.708    0.000    1.158    0.901
##   Gourmet_Food =~                                                            
##     Im10                   1.000                               0.810    0.921
##     Im14                   1.019    0.036   28.092    0.000    0.826    0.954
##   Trendiness =~                                                              
##     Im17                   1.000                               1.213    0.976
##     Im18                   0.979    0.042   23.480    0.000    1.188    0.850
##   Professionalism =~                                                         
##     Im16                   1.000                               0.920    0.765
##     Im19                   1.048    0.061   17.122    0.000    0.964    0.857
## 
## Covariances:
##                         Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Store_Decoration ~~                                                        
##     Luxury_Brands          0.528    0.063    8.353    0.000    0.462    0.462
##     French_Culture         0.410    0.064    6.433    0.000    0.338    0.338
##     Prdct_Assrtmnt         0.708    0.079    9.014    0.000    0.441    0.441
##     Gourmet_Food           0.416    0.050    8.377    0.000    0.416    0.416
##     Trendiness             0.768    0.076   10.093    0.000    0.512    0.512
##     Professionalsm         0.741    0.071   10.438    0.000    0.652    0.652
##   Luxury_Brands ~~                                                           
##     French_Culture         0.259    0.048    5.395    0.000    0.285    0.285
##     Prdct_Assrtmnt         0.589    0.066    8.916    0.000    0.489    0.489
##     Gourmet_Food           0.310    0.042    7.396    0.000    0.413    0.413
##     Trendiness             0.649    0.064   10.076    0.000    0.578    0.578
##     Professionalsm         0.441    0.054    8.217    0.000    0.517    0.517
##   French_Culture ~~                                                          
##     Prdct_Assrtmnt         0.291    0.061    4.772    0.000    0.227    0.227
##     Gourmet_Food           0.468    0.047    9.919    0.000    0.587    0.587
##     Trendiness             0.387    0.062    6.256    0.000    0.324    0.324
##     Professionalsm         0.334    0.051    6.494    0.000    0.369    0.369
##   Product_Assortment ~~                                                      
##     Gourmet_Food           0.328    0.050    6.598    0.000    0.311    0.311
##     Trendiness             0.817    0.079   10.368    0.000    0.517    0.517
##     Professionalsm         0.716    0.072    9.944    0.000    0.598    0.598
##   Gourmet_Food ~~                                                            
##     Trendiness             0.319    0.047    6.825    0.000    0.325    0.325
##     Professionalsm         0.371    0.043    8.559    0.000    0.497    0.497
##   Trendiness ~~                                                              
##     Professionalsm         0.669    0.066   10.060    0.000    0.599    0.599
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               4.995    0.056   88.555    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.994    0.000    4.999    3.713
##    .Im5               5.036    0.057   87.849    0.000    5.036    3.787
##    .Im12              5.665    0.049  116.085    0.000    5.665    4.988
##    .Im13              5.449    0.052  105.564    0.000    5.449    4.521
##    .Im6               5.827    0.051  113.782    0.000    5.827    4.858
##    .Im7               5.752    0.052  110.741    0.000    5.752    4.753
##    .Im1               4.792    0.057   84.219    0.000    4.792    3.598
##    .Im2               4.859    0.055   88.409    0.000    4.859    3.781
##    .Im10              6.100    0.037  162.785    0.000    6.100    6.937
##    .Im14              6.138    0.037  165.854    0.000    6.138    7.093
##    .Im17              5.027    0.053   94.537    0.000    5.027    4.043
##    .Im18              4.597    0.060   76.490    0.000    4.597    3.289
##    .Im16              5.136    0.052   99.152    0.000    5.136    4.269
##    .Im19              5.145    0.048  106.958    0.000    5.145    4.574
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               0.217    0.025    8.786    0.000    0.217    0.125
##    .Im4               0.105    0.024    4.305    0.000    0.105    0.058
##    .Im5               0.748    0.049   15.226    0.000    0.748    0.423
##    .Im12              0.433    0.048    9.117    0.000    0.433    0.336
##    .Im13              0.228    0.058    3.919    0.000    0.228    0.157
##    .Im6               0.472    0.057    8.327    0.000    0.472    0.328
##    .Im7               0.149    0.066    2.240    0.025    0.149    0.102
##    .Im1               0.079    0.051    1.533    0.125    0.079    0.044
##    .Im2               0.310    0.045    6.958    0.000    0.310    0.188
##    .Im10              0.117    0.020    5.971    0.000    0.117    0.151
##    .Im14              0.067    0.019    3.468    0.001    0.067    0.090
##    .Im17              0.074    0.048    1.559    0.119    0.074    0.048
##    .Im18              0.541    0.056    9.602    0.000    0.541    0.277
##    .Im16              0.601    0.052   11.506    0.000    0.601    0.415
##    .Im19              0.336    0.045    7.391    0.000    0.336    0.265
##     Store_Decoratn    1.524    0.107   14.289    0.000    1.000    1.000
##     Luxury_Brands     0.857    0.084   10.194    0.000    1.000    1.000
##     French_Culture    0.967    0.095   10.131    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.695    0.119   14.272    0.000    1.000    1.000
##     Gourmet_Food      0.657    0.050   13.252    0.000    1.000    1.000
##     Trendiness        1.472    0.105   14.013    0.000    1.000    1.000
##     Professionalsm    0.847    0.088    9.617    0.000    1.000    1.000

Modification indices of CFA Model with 7 Factors - Removing Im9+Im11 and Shopping_Experience Factor

modificationindices(fit_CFA) %>% filter(mi>10)
##            lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1 Gourmet_Food =~ Im12 11.162  0.185   0.150    0.132    0.132
## 2 Gourmet_Food =~ Im13 11.162 -0.221  -0.179   -0.149   -0.149
## 3 Gourmet_Food =~  Im6 10.546 -0.544  -0.441   -0.367   -0.367
## 4 Gourmet_Food =~  Im7 10.546  0.634   0.514    0.425    0.425
## 5          Im4 ~~ Im17 10.960 -0.050  -0.050   -0.567   -0.567

When we look at the modification indices of this CFA model we see that all the mi values are almost at 10. This is why we do not modify this model any further and stick with it.

Cronbachs alpha

We once again look at cronbachs alpha as we now have more and different data than in EFA. We hereby have to reassign the images to the factors according to the model.

Store_Decoration <- c("Im3","Im4","Im5")
Luxury_Brands <- c("Im12","Im13")
French_Culture <- c("Im6","Im7")
Product_Assortment <- c("Im1","Im2")
Gourmet_Food <- c("Im10","Im14")
Trendiness <- c("Im17","Im18")
Professionalism <- c("Im16","Im19")

Store Decoration

alpha.pa1 <- psych::alpha(df[Store_Decoration])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9156844 0.9156176 0.9000446 0.7834062 10.85081 0.006527817 5.009715 1.239067
##   median_r
##  0.7402258

raw_alpha is over 0.7 and average items correlation is above 0.5

Luxury Brands

alpha.pa1 <- psych::alpha(df[Luxury_Brands])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N       ase     mean       sd
##  0.8533477 0.8540858 0.7453314 0.7453314 5.853343 0.0124304 5.553114 1.097921
##   median_r
##  0.7453314

raw_alpha is over 0.7 and average items correlation is above 0.5

French Culture

alpha.pa1 <- psych::alpha(df[French_Culture])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.8788253 0.8788329 0.7838555 0.7838555 7.253067 0.01030533 5.792196 1.138545
##   median_r
##  0.7838555

raw_alpha is over 0.7 and average items correlation is above 0.5

Product Assortment

alpha.pa1 <- psych::alpha(df[Product_Assortment])
alpha.pa1$total
##  raw_alpha std.alpha  G6(smc) average_r      S/N         ase     mean       sd
##  0.9410747 0.9413295 0.889162  0.889162 16.04435 0.004999316 4.823315 1.271072
##  median_r
##  0.889162

raw_alpha is over 0.7 and average items correlation is above 0.5

Gourmet Food

alpha.pa1 <- psych::alpha(df[Gourmet_Food])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase    mean        sd
##  0.9231611 0.9234534 0.8577923 0.8577923 12.06394 0.006520538 6.11706 0.8503471
##   median_r
##  0.8577923

raw_alpha is over 0.7 and average items correlation is above 0.5

Trendiness

alpha.pa1 <- psych::alpha(df[Trendiness])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9058798 0.9094269 0.8338982 0.8338982 10.04081 0.007821878 4.812844 1.267096
##   median_r
##  0.8338982

raw_alpha is over 0.7 and average items correlation is above 0.5

Professionalism

alpha.pa1 <- psych::alpha(df[Professionalism])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.7981553 0.7993848 0.6658127 0.6658127 3.984668 0.01708793 5.141818 1.062645
##   median_r
##  0.6658127

raw_alpha is over 0.7 and average items correlation is above 0.5

Using the same evaluation methods as before we see that every factor seems to be consistent and relevant.

Correlation matrix

Here we look at the correlation matrix of the constructs

std_fit = inspect(fit_CFA, 'std')
std_fit$psi
##                    Str_Dc Lxry_B Frnc_C Prdc_A Grmt_F Trndns Prfssn
## Store_Decoration    1.000                                          
## Luxury_Brands       0.462  1.000                                   
## French_Culture      0.338  0.285  1.000                            
## Product_Assortment  0.441  0.489  0.227  1.000                     
## Gourmet_Food        0.416  0.413  0.587  0.311  1.000              
## Trendiness          0.512  0.578  0.324  0.517  0.325  1.000       
## Professionalism     0.652  0.517  0.369  0.598  0.497  0.599  1.000

We see that the biggest correlation is 0.652. Thereby there are no really big correlations.

Individual item Reliability

std.loadings<- inspect(fit_CFA, what="std")$lambda
check=std.loadings
check[check>0] <- 1
std.loadings[std.loadings==0] <- NA
std.loadings2 <- std.loadings^2
std.theta<- inspect(fit_CFA, what="std")$theta

#Individual item Reliability (should be larger than 0.4)
IIR=std.loadings2/(colSums(std.theta)+std.loadings2)
IIR
##      Str_Dc Lxry_B Frnc_C Prdc_A Grmt_F Trndns Prfssn
## Im3   0.875     NA     NA     NA     NA     NA     NA
## Im4   0.942     NA     NA     NA     NA     NA     NA
## Im5   0.577     NA     NA     NA     NA     NA     NA
## Im12     NA  0.664     NA     NA     NA     NA     NA
## Im13     NA  0.843     NA     NA     NA     NA     NA
## Im6      NA     NA  0.672     NA     NA     NA     NA
## Im7      NA     NA  0.898     NA     NA     NA     NA
## Im1      NA     NA     NA  0.956     NA     NA     NA
## Im2      NA     NA     NA  0.812     NA     NA     NA
## Im10     NA     NA     NA     NA  0.849     NA     NA
## Im14     NA     NA     NA     NA  0.910     NA     NA
## Im17     NA     NA     NA     NA     NA  0.952     NA
## Im18     NA     NA     NA     NA     NA  0.723     NA
## Im16     NA     NA     NA     NA     NA     NA  0.585
## Im19     NA     NA     NA     NA     NA     NA  0.735

We see that they are all above 0.4 which is the criterion they have to meet. Thereby each image expresses enough of the total variance of the factor.

Composite Construct Reliability

#Composite/Construct Reliability (should be higher than 0.6)
sum.std.loadings<-colSums(std.loadings, na.rm=TRUE)^2
sum.std.theta<-rowSums(std.theta)
sum.std.theta=check*sum.std.theta
CR=sum.std.loadings/(sum.std.loadings+colSums(sum.std.theta))
CR
##   Store_Decoration      Luxury_Brands     French_Culture Product_Assortment 
##          0.9214805          0.8591514          0.8791333          0.9382603 
##       Gourmet_Food         Trendiness    Professionalism 
##          0.9360069          0.9111139          0.7945510

Again this output is satisfying as every composite construct reliability is above 0.6.

Average Variance Extracted

#Average Variance Extracted (should be higher than 0.5)
std.loadings<- inspect(fit_CFA, what="std")$lambda
std.loadings <- std.loadings^2
AVE=colSums(std.loadings)/(colSums(sum.std.theta)+colSums(std.loadings))
AVE
##   Store_Decoration      Luxury_Brands     French_Culture Product_Assortment 
##          0.7981529          0.7537404          0.7852176          0.8838707 
##       Gourmet_Food         Trendiness    Professionalism 
##          0.8797436          0.8373822          0.6598574

Also finally this output is also good as all the average variance extracted for each factor is above 0.5

CFA Visualization

Model without Covariances

lavaanPlot(model = fit_CFA, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "palegreen4"), coefs = TRUE, sig = 0.05, covs = FALSE, digits = 2)

Model with Covariances

lavaanPlot(model = fit_CFA, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "palegreen4"), coefs = TRUE,sig = 0.05, covs = TRUE, digits = 2)

Structure Equation Modelling

Summary of SEM Model with 8 Factors

model_SEM1 <- "

# LATENT VARIABLES

## FACTORS CONSTRUCT

Shopping_Experience =~ Im20+Im21+Im22
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im11+Im12+Im13
French_Culture =~ Im6+Im7+Im9
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19

## MEDIATORS CONSTRUCT

Consumer_Satisfaction =~ SAT_1 + SAT_2 + SAT_3
Affective_Commitment =~ COM_A1 + COM_A2 + COM_A3 + COM_A4

## OUTCOMES CONSTRUCT

Repurchase_Intention =~ C_REP1 + C_REP2 + C_REP3
Cocreation_Intention =~ C_CR1 + C_CR3 + C_CR4

# REGRESSION

## DIRECT EFFECT

### CONSTRUCTS -> OUTCOMES

Repurchase_Intention ~ c11*Shopping_Experience + c12*Store_Decoration + c13*Luxury_Brands + c14*French_Culture + c15*Product_Assortment +c16*Gourmet_Food + c17*Trendiness + c18*Professionalism

Cocreation_Intention ~ c21*Shopping_Experience + c22*Store_Decoration + c23*Luxury_Brands + c24*French_Culture + c25*Product_Assortment +c26*Gourmet_Food + c27*Trendiness + c28*Professionalism

## MEDIATOR

### CONSTRUCTS -> MEDIATORS

Consumer_Satisfaction ~ a11*Shopping_Experience + a12*Store_Decoration + a13*Luxury_Brands + a14*French_Culture + a15*Product_Assortment +a16*Gourmet_Food + a17*Trendiness + a18*Professionalism

Affective_Commitment  ~ a21*Shopping_Experience + a22*Store_Decoration + a23*Luxury_Brands + a24*French_Culture + a25*Product_Assortment +a26*Gourmet_Food + a27*Trendiness + a28*Professionalism

### MEDIATORS ON OUTCOMES

Repurchase_Intention ~ b11*Consumer_Satisfaction
Repurchase_Intention ~ b12*Affective_Commitment

Cocreation_Intention ~ b21*Consumer_Satisfaction
Cocreation_Intention ~ b22*Affective_Commitment

## INDIRECT EFFECT

### REPURCHASE INTENTION

a11_b11 := a11*b11
a21_b12 := a21*b12

a12_b11 := a12*b11
a22_b12 := a22*b12

a13_b11 := a13*b11
a23_b12 := a23*b12

a14_b11 := a14*b11
a24_b12 := a24*b12

a15_b11 := a15*b11
a25_b12 := a25*b12

a16_b11 := a16*b11
a26_b12 := a26*b12

a17_b11 := a17*b11
a27_b12 := a27*b12

a18_b11 := a18*b11
a28_b12 := a28*b12

### COCREATION INTENTION

a11_b21 := a11*b21
a21_b22 := a21*b22

a12_b21 := a12*b21
a22_b22 := a22*b22

a13_b21 := a13*b21
a23_b22 := a23*b22

a14_b21 := a14*b21
a24_b22 := a24*b22

a15_b21 := a15*b21
a25_b22 := a25*b22

a16_b21 := a16*b21
a26_b22 := a26*b22

a17_b21 := a17*b21
a27_b22 := a27*b22

a18_b21 := a18*b21
a28_b22 := a28*b22

## TOTAL EFFECT

### REPURCHASE INTENTION

RI_Total_Shopping_Experience := c11 + (a11*b11) + (a21*b12)
RI_Total_Store_Decoration := c12 + (a12*b11) + (a22*b12)
RI_Total_Luxury_Brands := c13 + (a13*b11) + (a23*b12)
RI_Total_French_Culture := c14 + (a14*b11) + (a24*b12)
RI_Total_Product_Assortment := c15 + (a15*b11) + (a25*b12)
RI_Total_Gourmet_Food := c16 + (a16*b11) + (a26*b12)
RI_Total_Trendiness := c17 + (a17*b11) + (a27*b12)
RI_Total_Professionalism := c18 + (a18*b11) + (a28*b12)

### COCREATION INTENTION

CI_Total_Shopping_Experience := c21 + (a11*b21) + (a21*b22)
CI_Total_Store_Decoration := c22 + (a12*b21) + (a22*b22)
CI_Total_Luxury_Brands := c23 + (a13*b21) + (a23*b22)
CI_Total_French_Culture := c24 + (a14*b21) + (a24*b22)
CI_Total_Product_Assortment := c25 + (a15*b21) + (a25*b22)
CI_Total_Gourmet_Food := c26 + (a16*b21) + (a26*b22)
CI_Total_Trendiness := c27 + (a17*b21) + (a27*b22)
CI_Total_Professionalism := c28 + (a18*b21) + (a28*b22)


"


fit_SEM <- lavaan::sem(model_SEM1, data=df, missing="ML")

summary(fit_SEM,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 153 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                       164
## 
##   Number of observations                           553
##   Number of missing patterns                       137
## 
## Model Test User Model:
##                                                       
##   Test statistic                               835.621
##   Degrees of freedom                               430
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             12305.018
##   Degrees of freedom                               528
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.966
##   Tucker-Lewis Index (TLI)                       0.958
##                                                       
##   Robust Comparative Fit Index (CFI)             0.966
##   Robust Tucker-Lewis Index (TLI)                0.958
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -23197.252
##   Loglikelihood unrestricted model (H1)     -22779.442
##                                                       
##   Akaike (AIC)                               46722.504
##   Bayesian (BIC)                             47430.223
##   Sample-size adjusted Bayesian (SABIC)      46909.614
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.041
##   90 Percent confidence interval - lower         0.037
##   90 Percent confidence interval - upper         0.045
##   P-value H_0: RMSEA <= 0.050                    1.000
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.042
##   90 Percent confidence interval - lower         0.038
##   90 Percent confidence interval - upper         0.046
##   P-value H_0: Robust RMSEA <= 0.050             0.999
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.048
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                            Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   Shopping_Experience =~                                               
##     Im20                      1.000                               1.262
##     Im21                      0.857    0.041   20.998    0.000    1.081
##     Im22                      1.056    0.046   23.027    0.000    1.333
##   Store_Decoration =~                                                  
##     Im3                       1.000                               1.235
##     Im4                       1.057    0.025   42.733    0.000    1.306
##     Im5                       0.818    0.034   23.805    0.000    1.010
##   Luxury_Brands =~                                                     
##     Im11                      1.000                               0.700
##     Im12                      1.415    0.094   15.007    0.000    0.991
##     Im13                      1.468    0.105   13.929    0.000    1.029
##   French_Culture =~                                                    
##     Im6                       1.000                               1.004
##     Im7                       1.101    0.049   22.598    0.000    1.106
##     Im9                       0.788    0.057   13.939    0.000    0.792
##   Product_Assortment =~                                                
##     Im1                       1.000                               1.297
##     Im2                       0.895    0.032   28.321    0.000    1.161
##   Gourmet_Food =~                                                      
##     Im10                      1.000                               0.811
##     Im14                      1.018    0.035   28.748    0.000    0.826
##   Trendiness =~                                                        
##     Im17                      1.000                               1.205
##     Im18                      0.993    0.041   24.204    0.000    1.196
##   Professionalism =~                                                   
##     Im16                      1.000                               0.919
##     Im19                      1.043    0.058   17.879    0.000    0.959
##   Consumer_Satisfaction =~                                             
##     SAT_1                     1.000                               0.882
##     SAT_2                     0.933    0.049   18.916    0.000    0.823
##     SAT_3                     0.809    0.055   14.802    0.000    0.714
##   Affective_Commitment =~                                              
##     COM_A1                    1.000                               1.144
##     COM_A2                    1.174    0.055   21.504    0.000    1.343
##     COM_A3                    1.162    0.058   20.027    0.000    1.329
##     COM_A4                    1.278    0.061   20.800    0.000    1.462
##   Repurchase_Intention =~                                              
##     C_REP1                    1.000                               0.596
##     C_REP2                    0.971    0.043   22.489    0.000    0.579
##     C_REP3                    0.702    0.037   19.036    0.000    0.419
##   Cocreation_Intention =~                                              
##     C_CR1                     1.000                               1.658
##     C_CR3                     1.033    0.051   20.247    0.000    1.712
##     C_CR4                     0.964    0.049   19.766    0.000    1.598
##   Std.all
##          
##     0.844
##     0.789
##     0.873
##          
##     0.936
##     0.970
##     0.760
##          
##     0.613
##     0.872
##     0.855
##          
##     0.837
##     0.916
##     0.586
##          
##     0.974
##     0.904
##          
##     0.922
##     0.954
##          
##     0.969
##     0.856
##          
##     0.764
##     0.853
##          
##     0.865
##     0.819
##     0.624
##          
##     0.796
##     0.836
##     0.817
##     0.842
##          
##     0.816
##     0.931
##     0.756
##          
##     0.851
##     0.826
##     0.806
## 
## Regressions:
##                           Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Repurchase_Intention ~                                                       
##     Shppng_E (c11)           0.040    0.028    1.435    0.151    0.086    0.086
##     Str_Dcrt (c12)           0.010    0.029    0.353    0.724    0.021    0.021
##     Lxry_Brn (c13)           0.078    0.051    1.523    0.128    0.092    0.092
##     Frnch_Cl (c14)          -0.041    0.034   -1.187    0.235   -0.069   -0.069
##     Prdct_As (c15)          -0.017    0.026   -0.675    0.500   -0.038   -0.038
##     Gormt_Fd (c16)           0.042    0.044    0.963    0.335    0.057    0.057
##     Trendnss (c17)          -0.009    0.030   -0.295    0.768   -0.018   -0.018
##     Prfssnls (c18)          -0.037    0.060   -0.621    0.535   -0.058   -0.058
##   Cocreation_Intention ~                                                       
##     Shppng_E (c21)           0.152    0.087    1.738    0.082    0.116    0.116
##     Str_Dcrt (c22)          -0.030    0.090   -0.331    0.741   -0.022   -0.022
##     Lxry_Brn (c23)           0.201    0.159    1.264    0.206    0.085    0.085
##     Frnch_Cl (c24)          -0.134    0.107   -1.254    0.210   -0.081   -0.081
##     Prdct_As (c25)          -0.007    0.080   -0.091    0.927   -0.006   -0.006
##     Gormt_Fd (c26)          -0.074    0.136   -0.542    0.588   -0.036   -0.036
##     Trendnss (c27)           0.026    0.093    0.286    0.775    0.019    0.019
##     Prfssnls (c28)          -0.178    0.184   -0.967    0.334   -0.099   -0.099
##   Consumer_Satisfaction ~                                                      
##     Shppng_E (a11)           0.051    0.038    1.357    0.175    0.074    0.074
##     Str_Dcrt (a12)          -0.110    0.043   -2.551    0.011   -0.153   -0.153
##     Lxry_Brn (a13)          -0.041    0.075   -0.543    0.587   -0.032   -0.032
##     Frnch_Cl (a14)           0.109    0.050    2.156    0.031    0.124    0.124
##     Prdct_As (a15)           0.135    0.040    3.403    0.001    0.198    0.198
##     Gormt_Fd (a16)           0.075    0.066    1.147    0.251    0.069    0.069
##     Trendnss (a17)           0.004    0.045    0.089    0.929    0.005    0.005
##     Prfssnls (a18)           0.461    0.088    5.259    0.000    0.480    0.480
##   Affective_Commitment ~                                                       
##     Shppng_E (a21)           0.372    0.052    7.186    0.000    0.410    0.410
##     Str_Dcrt (a22)          -0.026    0.054   -0.480    0.631   -0.028   -0.028
##     Lxry_Brn (a23)          -0.193    0.098   -1.959    0.050   -0.118   -0.118
##     Frnch_Cl (a24)           0.237    0.065    3.621    0.000    0.208    0.208
##     Prdct_As (a25)           0.102    0.050    2.041    0.041    0.116    0.116
##     Gormt_Fd (a26)           0.016    0.085    0.194    0.846    0.012    0.012
##     Trendnss (a27)          -0.026    0.058   -0.450    0.653   -0.028   -0.028
##     Prfssnls (a28)           0.162    0.105    1.541    0.123    0.131    0.131
##   Repurchase_Intention ~                                                       
##     Cnsmr_St (b11)           0.215    0.045    4.785    0.000    0.318    0.318
##     Affctv_C (b12)           0.186    0.030    6.164    0.000    0.356    0.356
##   Cocreation_Intention ~                                                       
##     Cnsmr_St (b21)          -0.356    0.131   -2.710    0.007   -0.190   -0.190
##     Affctv_C (b22)           0.548    0.091    6.021    0.000    0.378    0.378
## 
## Covariances:
##                           Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                       
##     Store_Decoratn           0.728    0.082    8.916    0.000    0.467    0.467
##     Luxury_Brands            0.370    0.053    7.012    0.000    0.418    0.418
##     French_Culture           0.446    0.066    6.743    0.000    0.352    0.352
##     Prdct_Assrtmnt           0.732    0.084    8.676    0.000    0.447    0.447
##     Gourmet_Food             0.302    0.051    5.952    0.000    0.295    0.295
##     Trendiness               0.784    0.081    9.709    0.000    0.516    0.516
##     Professionalsm           0.552    0.068    8.107    0.000    0.476    0.476
##   Store_Decoration ~~                                                          
##     Luxury_Brands            0.407    0.051    8.023    0.000    0.470    0.470
##     French_Culture           0.452    0.063    7.146    0.000    0.364    0.364
##     Prdct_Assrtmnt           0.708    0.079    9.017    0.000    0.442    0.442
##     Gourmet_Food             0.417    0.050    8.396    0.000    0.417    0.417
##     Trendiness               0.769    0.076   10.130    0.000    0.516    0.516
##     Professionalsm           0.744    0.070   10.552    0.000    0.655    0.655
##   Luxury_Brands ~~                                                             
##     French_Culture           0.239    0.039    6.077    0.000    0.339    0.339
##     Prdct_Assrtmnt           0.433    0.053    8.112    0.000    0.477    0.477
##     Gourmet_Food             0.257    0.034    7.648    0.000    0.452    0.452
##     Trendiness               0.477    0.053    9.027    0.000    0.565    0.565
##     Professionalsm           0.342    0.043    7.967    0.000    0.531    0.531
##   French_Culture ~~                                                            
##     Prdct_Assrtmnt           0.323    0.063    5.160    0.000    0.248    0.248
##     Gourmet_Food             0.490    0.047   10.536    0.000    0.602    0.602
##     Trendiness               0.443    0.062    7.144    0.000    0.366    0.366
##     Professionalsm           0.362    0.052    6.979    0.000    0.392    0.392
##   Product_Assortment ~~                                                        
##     Gourmet_Food             0.328    0.050    6.606    0.000    0.312    0.312
##     Trendiness               0.814    0.079   10.355    0.000    0.521    0.521
##     Professionalsm           0.717    0.071   10.051    0.000    0.602    0.602
##   Gourmet_Food ~~                                                              
##     Trendiness               0.317    0.047    6.800    0.000    0.325    0.325
##     Professionalsm           0.372    0.043    8.640    0.000    0.500    0.500
##   Trendiness ~~                                                                
##     Professionalsm           0.667    0.066   10.108    0.000    0.602    0.602
##  .Repurchase_Intention ~~                                                      
##    .Cocretn_Intntn          -0.015    0.038   -0.404    0.686   -0.021   -0.021
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.218    0.000    4.672    3.125
##    .Im21              5.139    0.058   87.977    0.000    5.139    3.750
##    .Im22              4.280    0.065   65.479    0.000    4.280    2.802
##    .Im3               4.995    0.056   88.571    0.000    4.995    3.786
##    .Im4               4.999    0.057   87.000    0.000    4.999    3.713
##    .Im5               5.036    0.057   87.852    0.000    5.036    3.787
##    .Im11              5.653    0.049  115.303    0.000    5.653    4.944
##    .Im12              5.665    0.049  116.165    0.000    5.665    4.987
##    .Im13              5.448    0.052  105.695    0.000    5.448    4.528
##    .Im6               5.827    0.051  113.792    0.000    5.827    4.857
##    .Im7               5.752    0.052  111.063    0.000    5.752    4.765
##    .Im9               5.075    0.058   87.406    0.000    5.075    3.756
##    .Im1               4.792    0.057   84.290    0.000    4.792    3.600
##    .Im2               4.858    0.055   88.417    0.000    4.858    3.781
##    .Im10              6.100    0.037  162.786    0.000    6.100    6.936
##    .Im14              6.138    0.037  165.853    0.000    6.138    7.093
##    .Im17              5.025    0.053   94.560    0.000    5.025    4.043
##    .Im18              4.595    0.060   76.466    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.194    0.000    5.135    4.270
##    .Im19              5.145    0.048  107.020    0.000    5.145    4.576
##    .SAT_1             5.343    0.043  122.950    0.000    5.343    5.239
##    .SAT_2             5.482    0.043  127.738    0.000    5.482    5.455
##    .SAT_3             5.458    0.050  109.430    0.000    5.458    4.774
##    .COM_A1            4.287    0.061   69.747    0.000    4.287    2.983
##    .COM_A2            3.887    0.069   56.667    0.000    3.887    2.420
##    .COM_A3            3.543    0.070   50.857    0.000    3.543    2.178
##    .COM_A4            3.456    0.074   46.672    0.000    3.456    1.991
##    .C_REP1            4.283    0.031  137.513    0.000    4.283    5.859
##    .C_REP2            4.507    0.027  169.648    0.000    4.507    7.250
##    .C_REP3            4.677    0.024  196.940    0.000    4.677    8.445
##    .C_CR1             2.679    0.084   32.075    0.000    2.679    1.375
##    .C_CR3             3.261    0.088   36.880    0.000    3.261    1.572
##    .C_CR4             2.786    0.085   32.902    0.000    2.786    1.405
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
##    .Consmr_Stsfctn    0.000                               0.000    0.000
##    .Affctv_Cmmtmnt    0.000                               0.000    0.000
##    .Reprchs_Intntn    0.000                               0.000    0.000
##    .Cocretn_Intntn    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.644    0.059   10.840    0.000    0.644    0.288
##    .Im21              0.708    0.056   12.626    0.000    0.708    0.377
##    .Im22              0.557    0.061    9.070    0.000    0.557    0.239
##    .Im3               0.214    0.024    8.796    0.000    0.214    0.123
##    .Im4               0.108    0.024    4.485    0.000    0.108    0.059
##    .Im5               0.747    0.049   15.220    0.000    0.747    0.423
##    .Im11              0.817    0.055   14.817    0.000    0.817    0.625
##    .Im12              0.309    0.040    7.805    0.000    0.309    0.239
##    .Im13              0.390    0.045    8.754    0.000    0.390    0.269
##    .Im6               0.431    0.041   10.534    0.000    0.431    0.300
##    .Im7               0.234    0.041    5.682    0.000    0.234    0.161
##    .Im9               1.199    0.080   15.053    0.000    1.199    0.657
##    .Im1               0.089    0.047    1.918    0.055    0.089    0.050
##    .Im2               0.302    0.041    7.314    0.000    0.302    0.183
##    .Im10              0.116    0.019    6.156    0.000    0.116    0.150
##    .Im14              0.067    0.019    3.618    0.000    0.067    0.090
##    .Im17              0.094    0.045    2.085    0.037    0.094    0.061
##    .Im18              0.523    0.054    9.593    0.000    0.523    0.268
##    .Im16              0.602    0.050   11.943    0.000    0.602    0.416
##    .Im19              0.345    0.043    7.943    0.000    0.345    0.273
##    .SAT_1             0.262    0.034    7.733    0.000    0.262    0.252
##    .SAT_2             0.333    0.033    9.973    0.000    0.333    0.329
##    .SAT_3             0.798    0.056   14.348    0.000    0.798    0.610
##    .COM_A1            0.757    0.058   12.960    0.000    0.757    0.367
##    .COM_A2            0.778    0.065   11.906    0.000    0.778    0.301
##    .COM_A3            0.881    0.070   12.504    0.000    0.881    0.333
##    .COM_A4            0.876    0.075   11.720    0.000    0.876    0.291
##    .C_REP1            0.179    0.016   11.295    0.000    0.179    0.335
##    .C_REP2            0.051    0.010    4.947    0.000    0.051    0.133
##    .C_REP3            0.131    0.009   14.061    0.000    0.131    0.429
##    .C_CR1             1.048    0.113    9.309    0.000    1.048    0.276
##    .C_CR3             1.369    0.130   10.572    0.000    1.369    0.318
##    .C_CR4             1.377    0.122   11.292    0.000    1.377    0.350
##     Shoppng_Exprnc    1.591    0.136   11.660    0.000    1.000    1.000
##     Store_Decoratn    1.526    0.107   14.319    0.000    1.000    1.000
##     Luxury_Brands     0.491    0.067    7.337    0.000    1.000    1.000
##     French_Culture    1.008    0.089   11.380    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.683    0.117   14.435    0.000    1.000    1.000
##     Gourmet_Food      0.657    0.049   13.327    0.000    1.000    1.000
##     Trendiness        1.451    0.104   14.017    0.000    1.000    1.000
##     Professionalsm    0.845    0.087    9.730    0.000    1.000    1.000
##    .Consmr_Stsfctn    0.448    0.047    9.459    0.000    0.576    0.576
##    .Affctv_Cmmtmnt    0.859    0.086   10.029    0.000    0.657    0.657
##    .Reprchs_Intntn    0.237    0.022   10.939    0.000    0.667    0.667
##    .Cocretn_Intntn    2.278    0.208   10.939    0.000    0.829    0.829
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     a11_b11           0.011    0.008    1.314    0.189    0.023    0.023
##     a21_b12           0.069    0.014    4.797    0.000    0.146    0.146
##     a12_b11          -0.024    0.011   -2.239    0.025   -0.049   -0.049
##     a22_b12          -0.005    0.010   -0.479    0.632   -0.010   -0.010
##     a13_b11          -0.009    0.016   -0.539    0.590   -0.010   -0.010
##     a23_b12          -0.036    0.019   -1.863    0.062   -0.042   -0.042
##     a14_b11           0.023    0.012    1.976    0.048    0.039    0.039
##     a24_b12           0.044    0.014    3.130    0.002    0.074    0.074
##     a15_b11           0.029    0.010    2.853    0.004    0.063    0.063
##     a25_b12           0.019    0.010    1.943    0.052    0.041    0.041
##     a16_b11           0.016    0.014    1.119    0.263    0.022    0.022
##     a26_b12           0.003    0.016    0.194    0.846    0.004    0.004
##     a17_b11           0.001    0.010    0.089    0.929    0.002    0.002
##     a27_b12          -0.005    0.011   -0.449    0.653   -0.010   -0.010
##     a18_b11           0.099    0.028    3.486    0.000    0.153    0.153
##     a28_b12           0.030    0.020    1.495    0.135    0.046    0.046
##     a11_b21          -0.018    0.015   -1.203    0.229   -0.014   -0.014
##     a21_b22           0.204    0.043    4.754    0.000    0.155    0.155
##     a12_b21           0.039    0.021    1.868    0.062    0.029    0.029
##     a22_b22          -0.014    0.030   -0.478    0.632   -0.011   -0.011
##     a13_b21           0.015    0.027    0.534    0.594    0.006    0.006
##     a23_b22          -0.106    0.057   -1.858    0.063   -0.045   -0.045
##     a14_b21          -0.039    0.023   -1.700    0.089   -0.023   -0.023
##     a24_b22           0.130    0.042    3.114    0.002    0.079    0.079
##     a15_b21          -0.048    0.023   -2.113    0.035   -0.038   -0.038
##     a25_b22           0.056    0.029    1.948    0.051    0.044    0.044
##     a16_b21          -0.027    0.026   -1.053    0.293   -0.013   -0.013
##     a26_b22           0.009    0.046    0.194    0.846    0.004    0.004
##     a17_b21          -0.001    0.016   -0.089    0.929   -0.001   -0.001
##     a27_b22          -0.014    0.032   -0.449    0.654   -0.010   -0.010
##     a18_b21          -0.164    0.067   -2.441    0.015   -0.091   -0.091
##     a28_b22           0.089    0.060    1.483    0.138    0.049    0.049
##     RI_Ttl_Shppn_E    0.120    0.028    4.265    0.000    0.255    0.255
##     RI_Ttl_Str_Dcr   -0.018    0.030   -0.595    0.552   -0.038   -0.038
##     RI_Ttl_Lxry_Br    0.033    0.055    0.612    0.540    0.039    0.039
##     RI_Ttl_Frnch_C    0.026    0.036    0.734    0.463    0.045    0.045
##     RI_Ttl_Prdct_A    0.031    0.028    1.107    0.268    0.066    0.066
##     RI_Ttl_Grmt_Fd    0.062    0.047    1.298    0.194    0.084    0.084
##     RI_Ttl_Trndnss   -0.013    0.032   -0.396    0.692   -0.026   -0.026
##     RI_Ttl_Prfssnl    0.092    0.059    1.569    0.117    0.142    0.142
##     CI_Ttl_Shppn_E    0.337    0.084    4.024    0.000    0.257    0.257
##     CI_Ttl_Str_Dcr   -0.005    0.092   -0.056    0.955   -0.004   -0.004
##     CI_Ttl_Lxry_Br    0.109    0.164    0.668    0.504    0.046    0.046
##     CI_Ttl_Frnch_C   -0.043    0.109   -0.397    0.691   -0.026   -0.026
##     CI_Ttl_Prdct_A    0.001    0.083    0.007    0.995    0.000    0.000
##     CI_Ttl_Grmt_Fd   -0.091    0.142   -0.644    0.520   -0.045   -0.045
##     CI_Ttl_Trndnss    0.011    0.097    0.111    0.912    0.008    0.008
##     CI_Ttl_Prfssnl   -0.253    0.172   -1.471    0.141   -0.141   -0.141

Model Criterions

RMSEA: 0.041

Akaike (AIC): 46722.504

Bayesian (BIC): 47430.223

Comparative Fit Index (CFI): 0.966

Tucker-Lewis Index (TLI): 0.958

Overall good fitting of this model

Significant Coefficients at 10% alpha level

parameterestimates(fit_SEM, standardized = TRUE) %>% select(lhs, rhs, est, pvalue,ci.lower, ci.upper) %>% filter(pvalue < 0.1) %>% filter(row_number() > 140)
##                             lhs                     rhs    est pvalue ci.lower
## 1                       a21_b12                 a21*b12  0.069  0.000    0.041
## 2                       a12_b11                 a12*b11 -0.024  0.025   -0.044
## 3                       a23_b12                 a23*b12 -0.036  0.062   -0.073
## 4                       a14_b11                 a14*b11  0.023  0.048    0.000
## 5                       a24_b12                 a24*b12  0.044  0.002    0.016
## 6                       a15_b11                 a15*b11  0.029  0.004    0.009
## 7                       a25_b12                 a25*b12  0.019  0.052    0.000
## 8                       a18_b11                 a18*b11  0.099  0.000    0.043
## 9                       a21_b22                 a21*b22  0.204  0.000    0.120
## 10                      a12_b21                 a12*b21  0.039  0.062   -0.002
## 11                      a23_b22                 a23*b22 -0.106  0.063   -0.217
## 12                      a14_b21                 a14*b21 -0.039  0.089   -0.083
## 13                      a24_b22                 a24*b22  0.130  0.002    0.048
## 14                      a15_b21                 a15*b21 -0.048  0.035   -0.093
## 15                      a25_b22                 a25*b22  0.056  0.051    0.000
## 16                      a18_b21                 a18*b21 -0.164  0.015   -0.296
## 17 RI_Total_Shopping_Experience c11+(a11*b11)+(a21*b12)  0.120  0.000    0.065
## 18 CI_Total_Shopping_Experience c21+(a11*b21)+(a21*b22)  0.337  0.000    0.173
##    ci.upper
## 1     0.097
## 2    -0.003
## 3     0.002
## 4     0.047
## 5     0.071
## 6     0.049
## 7     0.038
## 8     0.155
## 9     0.288
## 10    0.080
## 11    0.006
## 12    0.006
## 13    0.212
## 14   -0.003
## 15    0.112
## 16   -0.032
## 17    0.176
## 18    0.501

Significant Coefficients at 5% alpha level

We will only account those coefficient and variables for this model.

parameterestimates(fit_SEM, standardized = TRUE) %>% select(lhs, rhs, est, pvalue,ci.lower, ci.upper) %>% filter(pvalue < 0.05) %>% filter(row_number() > 137)
##                             lhs                     rhs    est pvalue ci.lower
## 1                       a21_b12                 a21*b12  0.069  0.000    0.041
## 2                       a12_b11                 a12*b11 -0.024  0.025   -0.044
## 3                       a14_b11                 a14*b11  0.023  0.048    0.000
## 4                       a24_b12                 a24*b12  0.044  0.002    0.016
## 5                       a15_b11                 a15*b11  0.029  0.004    0.009
## 6                       a18_b11                 a18*b11  0.099  0.000    0.043
## 7                       a21_b22                 a21*b22  0.204  0.000    0.120
## 8                       a24_b22                 a24*b22  0.130  0.002    0.048
## 9                       a15_b21                 a15*b21 -0.048  0.035   -0.093
## 10                      a18_b21                 a18*b21 -0.164  0.015   -0.296
## 11 RI_Total_Shopping_Experience c11+(a11*b11)+(a21*b12)  0.120  0.000    0.065
## 12 CI_Total_Shopping_Experience c21+(a11*b21)+(a21*b22)  0.337  0.000    0.173
##    ci.upper
## 1     0.097
## 2    -0.003
## 3     0.047
## 4     0.071
## 5     0.049
## 6     0.155
## 7     0.288
## 8     0.212
## 9    -0.003
## 10   -0.032
## 11    0.176
## 12    0.501

Coefficient Interpretation

Indirect Effect

a21_b12: Shopping_Experience -> Affective_Commitment -> Repurchase_Intention (Positive)

a12_b11: Store_Decoration -> Consumer_Satisfaction -> Repurchase_Intention (Negative)

a14_b11: French_Culture -> Consumer_Satisfaction -> Repurchase_Intention (Positive)

a24_b12: French_Culture -> Affective_Commitment -> Repurchase_Intention (Positive)

a15_b11: Product_Assortment -> Consumer_Satisfaction -> Repurchase_Intention (Positive)

a18_b11: Professionalism -> Consumer_Satisfaction -> Repurchase_Intention (Positive)

a21_b22: Shopping_Experience -> Affective_Commitment -> Cocreation_Intention (Positive)

a24_b22: French_Culture -> Affective_Commitment -> Cocreation_Intention (Positive)

a15_b21: Product_Assortment -> Consumer_Satisfaction -> Cocreation_Intention (Negative)

a18_b21: Professionalism -> Consumer_Satisfaction -> Cocreation_Intention (Negative)

Total Effect

RI_Total_Shopping_Experience (Positive)

CI_Total_Shopping_Experience (Positive)

Summary of SEM Model with 7 Factors (Im9 + Im11 + Store_Experience Removed)

model_SEM2 <- "

# LATENT VARIABLES

## FACTORS CONSTRUCT

Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im12+Im13
French_Culture =~ Im6+Im7
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19

## MEDIATORS CONSTRUCT

Consumer_Satisfaction =~ SAT_1 + SAT_2 + SAT_3
Affective_Commitment =~ COM_A1 + COM_A2 + COM_A3 + COM_A4

## OUTCOMES CONSTRUCT

Repurchase_Intention =~ C_REP1 + C_REP2 + C_REP3
Cocreation_Intention =~ C_CR1 + C_CR3 + C_CR4

# REGRESSION

## DIRECT EFFECT

### CONSTRUCTS -> OUTCOMES

Repurchase_Intention ~ c12*Store_Decoration + c13*Luxury_Brands + c14*French_Culture + c15*Product_Assortment +c16*Gourmet_Food + c17*Trendiness + c18*Professionalism

Cocreation_Intention ~ c22*Store_Decoration + c23*Luxury_Brands + c24*French_Culture + c25*Product_Assortment +c26*Gourmet_Food + c27*Trendiness + c28*Professionalism

## MEDIATOR

### CONSTRUCTS -> MEDIATORS

Consumer_Satisfaction ~ a12*Store_Decoration + a13*Luxury_Brands + a14*French_Culture + a15*Product_Assortment +a16*Gourmet_Food + a17*Trendiness + a18*Professionalism

Affective_Commitment  ~ a22*Store_Decoration + a23*Luxury_Brands + a24*French_Culture + a25*Product_Assortment +a26*Gourmet_Food + a27*Trendiness + a28*Professionalism

### MEDIATORS ON OUTCOMES

Repurchase_Intention ~ b11*Consumer_Satisfaction
Repurchase_Intention ~ b12*Affective_Commitment

Cocreation_Intention ~ b21*Consumer_Satisfaction
Cocreation_Intention ~ b22*Affective_Commitment

## INDIRECT EFFECT

### REPURCHASE INTENTION

a12_b11 := a12*b11
a22_b12 := a22*b12

a13_b11 := a13*b11
a23_b12 := a23*b12

a14_b11 := a14*b11
a24_b12 := a24*b12

a15_b11 := a15*b11
a25_b12 := a25*b12

a16_b11 := a16*b11
a26_b12 := a26*b12

a17_b11 := a17*b11
a27_b12 := a27*b12

a18_b11 := a18*b11
a28_b12 := a28*b12

### COCREATION INTENTION

a12_b21 := a12*b21
a22_b22 := a22*b22

a13_b21 := a13*b21
a23_b22 := a23*b22

a14_b21 := a14*b21
a24_b22 := a24*b22

a15_b21 := a15*b21
a25_b22 := a25*b22

a16_b21 := a16*b21
a26_b22 := a26*b22

a17_b21 := a17*b21
a27_b22 := a27*b22

a18_b21 := a18*b21
a28_b22 := a28*b22

## TOTAL EFFECT

### REPURCHASE INTENTION

RI_Total_Store_Decoration := c12 + (a12*b11) + (a22*b12)
RI_Total_Luxury_Brands := c13 + (a13*b11) + (a23*b12)
RI_Total_French_Culture := c14 + (a14*b11) + (a24*b12)
RI_Total_Product_Assortment := c15 + (a15*b11) + (a25*b12)
RI_Total_Gourmet_Food := c16 + (a16*b11) + (a26*b12)
RI_Total_Trendiness := c17 + (a17*b11) + (a27*b12)
RI_Total_Professionalism := c18 + (a18*b11) + (a28*b12)

### COCREATION INTENTION

CI_Total_Store_Decoration := c22 + (a12*b21) + (a22*b22)
CI_Total_Luxury_Brands := c23 + (a13*b21) + (a23*b22)
CI_Total_French_Culture := c24 + (a14*b21) + (a24*b22)
CI_Total_Product_Assortment := c25 + (a15*b21) + (a25*b22)
CI_Total_Gourmet_Food := c26 + (a16*b21) + (a26*b22)
CI_Total_Trendiness := c27 + (a17*b21) + (a27*b21)
CI_Total_Professionalism := c28 + (a18*b21) + (a28*b22)


"


fit_SEM <- lavaan::sem(model_SEM2, data=df, missing="ML")

summary(fit_SEM,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 141 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                       138
## 
##   Number of observations                           553
##   Number of missing patterns                       121
## 
## Model Test User Model:
##                                                       
##   Test statistic                               499.545
##   Degrees of freedom                               296
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             10473.523
##   Degrees of freedom                               378
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.980
##   Tucker-Lewis Index (TLI)                       0.974
##                                                       
##   Robust Comparative Fit Index (CFI)             0.980
##   Robust Tucker-Lewis Index (TLI)                0.974
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -19250.075
##   Loglikelihood unrestricted model (H1)     -19000.303
##                                                       
##   Akaike (AIC)                               38776.151
##   Bayesian (BIC)                             39371.670
##   Sample-size adjusted Bayesian (SABIC)      38933.597
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.035
##   90 Percent confidence interval - lower         0.030
##   90 Percent confidence interval - upper         0.041
##   P-value H_0: RMSEA <= 0.050                    1.000
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.036
##   90 Percent confidence interval - lower         0.030
##   90 Percent confidence interval - upper         0.041
##   P-value H_0: Robust RMSEA <= 0.050             1.000
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.041
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                            Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   Store_Decoration =~                                                  
##     Im3                       1.000                               1.234
##     Im4                       1.059    0.025   42.561    0.000    1.307
##     Im5                       0.818    0.034   23.772    0.000    1.010
##   Luxury_Brands =~                                                     
##     Im12                      1.000                               0.927
##     Im13                      1.190    0.068   17.560    0.000    1.103
##   French_Culture =~                                                    
##     Im6                       1.000                               0.991
##     Im7                       1.147    0.064   18.044    0.000    1.138
##   Product_Assortment =~                                                
##     Im1                       1.000                               1.294
##     Im2                       0.899    0.031   28.566    0.000    1.164
##   Gourmet_Food =~                                                      
##     Im10                      1.000                               0.808
##     Im14                      1.024    0.036   28.262    0.000    0.828
##   Trendiness =~                                                        
##     Im17                      1.000                               1.215
##     Im18                      0.976    0.041   23.567    0.000    1.186
##   Professionalism =~                                                   
##     Im16                      1.000                               0.918
##     Im19                      1.044    0.058   17.850    0.000    0.959
##   Consumer_Satisfaction =~                                             
##     SAT_1                     1.000                               0.883
##     SAT_2                     0.932    0.049   18.898    0.000    0.822
##     SAT_3                     0.809    0.055   14.805    0.000    0.715
##   Affective_Commitment =~                                              
##     COM_A1                    1.000                               1.150
##     COM_A2                    1.169    0.054   21.629    0.000    1.344
##     COM_A3                    1.157    0.058   20.063    0.000    1.330
##     COM_A4                    1.265    0.061   20.760    0.000    1.454
##   Repurchase_Intention =~                                              
##     C_REP1                    1.000                               0.594
##     C_REP2                    0.975    0.043   22.499    0.000    0.579
##     C_REP3                    0.704    0.037   19.074    0.000    0.418
##   Cocreation_Intention =~                                              
##     C_CR1                     1.000                               1.663
##     C_CR3                     1.033    0.051   20.190    0.000    1.718
##     C_CR4                     0.960    0.049   19.742    0.000    1.597
##   Std.all
##          
##     0.935
##     0.971
##     0.759
##          
##     0.816
##     0.916
##          
##     0.827
##     0.940
##          
##     0.972
##     0.906
##          
##     0.919
##     0.956
##          
##     0.977
##     0.849
##          
##     0.763
##     0.853
##          
##     0.865
##     0.818
##     0.625
##          
##     0.800
##     0.837
##     0.817
##     0.838
##          
##     0.813
##     0.933
##     0.755
##          
##     0.852
##     0.827
##     0.804
## 
## Regressions:
##                           Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Repurchase_Intention ~                                                       
##     Str_Dcrt (c12)           0.015    0.029    0.526    0.599    0.032    0.032
##     Lxry_Brn (c13)           0.069    0.038    1.819    0.069    0.107    0.107
##     Frnch_Cl (c14)          -0.031    0.033   -0.922    0.357   -0.051   -0.051
##     Prdct_As (c15)          -0.015    0.026   -0.580    0.562   -0.033   -0.033
##     Gormt_Fd (c16)           0.037    0.043    0.852    0.394    0.050    0.050
##     Trendnss (c17)          -0.004    0.029   -0.147    0.883   -0.009   -0.009
##     Prfssnls (c18)          -0.032    0.060   -0.537    0.591   -0.050   -0.050
##   Cocreation_Intention ~                                                       
##     Str_Dcrt (c22)          -0.003    0.090   -0.039    0.969   -0.003   -0.003
##     Lxry_Brn (c23)           0.116    0.119    0.979    0.328    0.065    0.065
##     Frnch_Cl (c24)          -0.115    0.105   -1.104    0.270   -0.069   -0.069
##     Prdct_As (c25)           0.015    0.081    0.179    0.858    0.011    0.011
##     Gormt_Fd (c26)          -0.072    0.134   -0.538    0.591   -0.035   -0.035
##     Trendnss (c27)           0.061    0.090    0.678    0.498    0.044    0.044
##     Prfssnls (c28)          -0.159    0.186   -0.858    0.391   -0.088   -0.088
##   Consumer_Satisfaction ~                                                      
##     Str_Dcrt (a12)          -0.101    0.043   -2.367    0.018   -0.141   -0.141
##     Lxry_Brn (a13)          -0.017    0.056   -0.311    0.756   -0.018   -0.018
##     Frnch_Cl (a14)           0.115    0.049    2.380    0.017    0.130    0.130
##     Prdct_As (a15)           0.144    0.040    3.617    0.000    0.211    0.211
##     Gormt_Fd (a16)           0.070    0.064    1.090    0.276    0.064    0.064
##     Trendnss (a17)           0.020    0.043    0.453    0.650    0.027    0.027
##     Prfssnls (a18)           0.464    0.088    5.278    0.000    0.483    0.483
##   Affective_Commitment ~                                                       
##     Str_Dcrt (a22)           0.042    0.057    0.746    0.456    0.046    0.046
##     Lxry_Brn (a23)          -0.141    0.076   -1.846    0.065   -0.114   -0.114
##     Frnch_Cl (a24)           0.295    0.067    4.427    0.000    0.254    0.254
##     Prdct_As (a25)           0.168    0.053    3.178    0.001    0.189    0.189
##     Gormt_Fd (a26)          -0.004    0.088   -0.041    0.967   -0.003   -0.003
##     Trendnss (a27)           0.082    0.059    1.397    0.162    0.087    0.087
##     Prfssnls (a28)           0.198    0.112    1.772    0.076    0.158    0.158
##   Repurchase_Intention ~                                                       
##     Cnsmr_St (b11)           0.210    0.045    4.678    0.000    0.312    0.312
##     Affctv_C (b12)           0.201    0.028    7.198    0.000    0.389    0.389
##   Cocreation_Intention ~                                                       
##     Cnsmr_St (b21)          -0.368    0.132   -2.782    0.005   -0.195   -0.195
##     Affctv_C (b22)           0.601    0.084    7.155    0.000    0.415    0.415
## 
## Covariances:
##                           Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Store_Decoration ~~                                                          
##     Luxury_Brands            0.529    0.063    8.372    0.000    0.462    0.462
##     French_Culture           0.418    0.063    6.598    0.000    0.342    0.342
##     Prdct_Assrtmnt           0.706    0.078    8.999    0.000    0.442    0.442
##     Gourmet_Food             0.415    0.050    8.370    0.000    0.416    0.416
##     Trendiness               0.765    0.076   10.073    0.000    0.511    0.511
##     Professionalsm           0.742    0.070   10.537    0.000    0.655    0.655
##   Luxury_Brands ~~                                                             
##     French_Culture           0.261    0.048    5.409    0.000    0.285    0.285
##     Prdct_Assrtmnt           0.585    0.066    8.890    0.000    0.488    0.488
##     Gourmet_Food             0.310    0.042    7.416    0.000    0.414    0.414
##     Trendiness               0.648    0.064   10.102    0.000    0.576    0.576
##     Professionalsm           0.442    0.053    8.265    0.000    0.520    0.520
##   French_Culture ~~                                                            
##     Prdct_Assrtmnt           0.295    0.061    4.807    0.000    0.230    0.230
##     Gourmet_Food             0.471    0.047   10.087    0.000    0.588    0.588
##     Trendiness               0.394    0.061    6.411    0.000    0.327    0.327
##     Professionalsm           0.338    0.051    6.605    0.000    0.371    0.371
##   Product_Assortment ~~                                                        
##     Gourmet_Food             0.327    0.049    6.617    0.000    0.313    0.313
##     Trendiness               0.814    0.079   10.364    0.000    0.518    0.518
##     Professionalsm           0.716    0.071   10.043    0.000    0.603    0.603
##   Gourmet_Food ~~                                                              
##     Trendiness               0.318    0.047    6.807    0.000    0.324    0.324
##     Professionalsm           0.370    0.043    8.603    0.000    0.499    0.499
##   Trendiness ~~                                                                
##     Professionalsm           0.669    0.066   10.136    0.000    0.600    0.600
##  .Repurchase_Intention ~~                                                      
##    .Cocretn_Intntn          -0.008    0.039   -0.211    0.833   -0.011   -0.011
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               4.995    0.056   88.553    0.000    4.995    3.786
##    .Im4               4.998    0.057   87.002    0.000    4.998    3.713
##    .Im5               5.036    0.057   87.850    0.000    5.036    3.787
##    .Im12              5.665    0.049  116.144    0.000    5.665    4.990
##    .Im13              5.448    0.052  105.648    0.000    5.448    4.525
##    .Im6               5.827    0.051  113.798    0.000    5.827    4.858
##    .Im7               5.754    0.052  110.775    0.000    5.754    4.754
##    .Im1               4.792    0.057   84.306    0.000    4.792    3.600
##    .Im2               4.859    0.055   88.449    0.000    4.859    3.782
##    .Im10              6.100    0.037  162.786    0.000    6.100    6.936
##    .Im14              6.138    0.037  165.855    0.000    6.138    7.093
##    .Im17              5.026    0.053   94.592    0.000    5.026    4.044
##    .Im18              4.596    0.060   76.498    0.000    4.596    3.290
##    .Im16              5.135    0.052   99.198    0.000    5.135    4.270
##    .Im19              5.145    0.048  107.023    0.000    5.145    4.576
##    .SAT_1             5.343    0.043  122.942    0.000    5.343    5.238
##    .SAT_2             5.482    0.043  127.742    0.000    5.482    5.455
##    .SAT_3             5.458    0.050  109.425    0.000    5.458    4.773
##    .COM_A1            4.286    0.061   69.730    0.000    4.286    2.982
##    .COM_A2            3.886    0.069   56.659    0.000    3.886    2.419
##    .COM_A3            3.544    0.070   50.855    0.000    3.544    2.178
##    .COM_A4            3.455    0.074   46.646    0.000    3.455    1.990
##    .C_REP1            4.283    0.031  137.734    0.000    4.283    5.869
##    .C_REP2            4.507    0.027  169.982    0.000    4.507    7.264
##    .C_REP3            4.677    0.024  197.229    0.000    4.677    8.458
##    .C_CR1             2.678    0.084   32.018    0.000    2.678    1.372
##    .C_CR3             3.261    0.089   36.826    0.000    3.261    1.570
##    .C_CR4             2.786    0.085   32.858    0.000    2.786    1.403
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
##    .Consmr_Stsfctn    0.000                               0.000    0.000
##    .Affctv_Cmmtmnt    0.000                               0.000    0.000
##    .Reprchs_Intntn    0.000                               0.000    0.000
##    .Cocretn_Intntn    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               0.217    0.025    8.813    0.000    0.217    0.125
##    .Im4               0.104    0.024    4.285    0.000    0.104    0.057
##    .Im5               0.748    0.049   15.225    0.000    0.748    0.423
##    .Im12              0.430    0.047    9.082    0.000    0.430    0.334
##    .Im13              0.233    0.058    4.038    0.000    0.233    0.160
##    .Im6               0.456    0.052    8.709    0.000    0.456    0.317
##    .Im7               0.171    0.059    2.886    0.004    0.171    0.117
##    .Im1               0.097    0.046    2.116    0.034    0.097    0.055
##    .Im2               0.296    0.041    7.250    0.000    0.296    0.179
##    .Im10              0.120    0.019    6.187    0.000    0.120    0.155
##    .Im14              0.064    0.019    3.327    0.001    0.064    0.085
##    .Im17              0.069    0.047    1.462    0.144    0.069    0.045
##    .Im18              0.546    0.056    9.728    0.000    0.546    0.280
##    .Im16              0.603    0.050   11.968    0.000    0.603    0.417
##    .Im19              0.345    0.044    7.901    0.000    0.345    0.273
##    .SAT_1             0.261    0.034    7.705    0.000    0.261    0.251
##    .SAT_2             0.333    0.033    9.988    0.000    0.333    0.330
##    .SAT_3             0.797    0.056   14.344    0.000    0.797    0.610
##    .COM_A1            0.743    0.058   12.803    0.000    0.743    0.360
##    .COM_A2            0.774    0.065   11.826    0.000    0.774    0.300
##    .COM_A3            0.879    0.071   12.385    0.000    0.879    0.332
##    .COM_A4            0.899    0.076   11.840    0.000    0.899    0.298
##    .C_REP1            0.180    0.016   11.399    0.000    0.180    0.338
##    .C_REP2            0.050    0.010    4.846    0.000    0.050    0.130
##    .C_REP3            0.131    0.009   14.078    0.000    0.131    0.429
##    .C_CR1             1.042    0.113    9.225    0.000    1.042    0.274
##    .C_CR3             1.361    0.130   10.484    0.000    1.361    0.316
##    .C_CR4             1.390    0.122   11.392    0.000    1.390    0.353
##     Store_Decoratn    1.523    0.107   14.285    0.000    1.000    1.000
##     Luxury_Brands     0.859    0.084   10.238    0.000    1.000    1.000
##     French_Culture    0.983    0.094   10.507    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.675    0.116   14.408    0.000    1.000    1.000
##     Gourmet_Food      0.654    0.049   13.225    0.000    1.000    1.000
##     Trendiness        1.475    0.105   14.080    0.000    1.000    1.000
##     Professionalsm    0.843    0.087    9.717    0.000    1.000    1.000
##    .Consmr_Stsfctn    0.450    0.048    9.453    0.000    0.578    0.578
##    .Affctv_Cmmtmnt    1.006    0.096   10.435    0.000    0.761    0.761
##    .Reprchs_Intntn    0.238    0.022   10.913    0.000    0.676    0.676
##    .Cocretn_Intntn    2.311    0.211   10.964    0.000    0.835    0.835
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     a12_b11          -0.021    0.010   -2.099    0.036   -0.044   -0.044
##     a22_b12           0.009    0.011    0.742    0.458    0.018    0.018
##     a13_b11          -0.004    0.012   -0.310    0.757   -0.006   -0.006
##     a23_b12          -0.028    0.016   -1.784    0.074   -0.044   -0.044
##     a14_b11           0.024    0.011    2.138    0.032    0.040    0.040
##     a24_b12           0.059    0.016    3.803    0.000    0.099    0.099
##     a15_b11           0.030    0.010    2.948    0.003    0.066    0.066
##     a25_b12           0.034    0.012    2.926    0.003    0.073    0.073
##     a16_b11           0.015    0.014    1.064    0.287    0.020    0.020
##     a26_b12          -0.001    0.018   -0.041    0.967   -0.001   -0.001
##     a17_b11           0.004    0.009    0.451    0.652    0.008    0.008
##     a27_b12           0.016    0.012    1.372    0.170    0.034    0.034
##     a18_b11           0.097    0.028    3.455    0.001    0.151    0.151
##     a28_b12           0.040    0.023    1.723    0.085    0.062    0.062
##     a12_b21           0.037    0.020    1.815    0.070    0.028    0.028
##     a22_b22           0.026    0.034    0.744    0.457    0.019    0.019
##     a13_b21           0.006    0.021    0.309    0.757    0.004    0.004
##     a23_b22          -0.085    0.047   -1.786    0.074   -0.047   -0.047
##     a14_b21          -0.042    0.023   -1.817    0.069   -0.025   -0.025
##     a24_b22           0.177    0.047    3.802    0.000    0.106    0.106
##     a15_b21          -0.053    0.024   -2.196    0.028   -0.041   -0.041
##     a25_b22           0.101    0.034    2.946    0.003    0.078    0.078
##     a16_b21          -0.026    0.026   -1.012    0.312   -0.013   -0.013
##     a26_b22          -0.002    0.053   -0.041    0.967   -0.001   -0.001
##     a17_b21          -0.007    0.016   -0.447    0.655   -0.005   -0.005
##     a27_b22           0.049    0.036    1.376    0.169    0.036    0.036
##     a18_b21          -0.171    0.069   -2.490    0.013   -0.094   -0.094
##     a28_b22           0.119    0.070    1.712    0.087    0.066    0.066
##     RI_Ttl_Str_Dcr    0.003    0.030    0.084    0.933    0.005    0.005
##     RI_Ttl_Lxry_Br    0.037    0.041    0.896    0.370    0.057    0.057
##     RI_Ttl_Frnch_C    0.053    0.035    1.499    0.134    0.088    0.088
##     RI_Ttl_Prdct_A    0.049    0.028    1.742    0.081    0.106    0.106
##     RI_Ttl_Grmt_Fd    0.051    0.047    1.080    0.280    0.069    0.069
##     RI_Ttl_Trndnss    0.016    0.031    0.524    0.601    0.034    0.034
##     RI_Ttl_Prfssnl    0.105    0.059    1.761    0.078    0.162    0.162
##     CI_Ttl_Str_Dcr    0.059    0.092    0.640    0.522    0.044    0.044
##     CI_Ttl_Lxry_Br    0.038    0.124    0.305    0.760    0.021    0.021
##     CI_Ttl_Frnch_C    0.019    0.107    0.183    0.855    0.012    0.012
##     CI_Ttl_Prdct_A    0.062    0.085    0.738    0.460    0.049    0.049
##     CI_Ttl_Grmt_Fd   -0.100    0.141   -0.706    0.480   -0.049   -0.049
##     CI_Ttl_Trndnss    0.023    0.096    0.243    0.808    0.022    0.022
##     CI_Ttl_Prfssnl   -0.211    0.175   -1.204    0.229   -0.116   -0.116

Model Criterions

RMSEA: 0.035

Akaike (AIC): 38776.151

Bayesian (BIC): 39371.670

Comparative Fit Index (CFI): 0.980

Tucker-Lewis Index (TLI): 0.974

Overall good fitting of this model even better than the previous model with 8 factors

Significant Coefficients at 10% alpha level

parameterestimates(fit_SEM, standardized = TRUE) %>% select(lhs, rhs, est, pvalue,ci.lower, ci.upper) %>% filter(pvalue < 0.1) %>% filter(row_number() > 117)
##                            lhs                     rhs    est pvalue ci.lower
## 1                      a12_b11                 a12*b11 -0.021  0.036   -0.041
## 2                      a23_b12                 a23*b12 -0.028  0.074   -0.059
## 3                      a14_b11                 a14*b11  0.024  0.032    0.002
## 4                      a24_b12                 a24*b12  0.059  0.000    0.029
## 5                      a15_b11                 a15*b11  0.030  0.003    0.010
## 6                      a25_b12                 a25*b12  0.034  0.003    0.011
## 7                      a18_b11                 a18*b11  0.097  0.001    0.042
## 8                      a28_b12                 a28*b12  0.040  0.085   -0.005
## 9                      a12_b21                 a12*b21  0.037  0.070   -0.003
## 10                     a23_b22                 a23*b22 -0.085  0.074   -0.178
## 11                     a14_b21                 a14*b21 -0.042  0.069   -0.088
## 12                     a24_b22                 a24*b22  0.177  0.000    0.086
## 13                     a15_b21                 a15*b21 -0.053  0.028   -0.100
## 14                     a25_b22                 a25*b22  0.101  0.003    0.034
## 15                     a18_b21                 a18*b21 -0.171  0.013   -0.305
## 16                     a28_b22                 a28*b22  0.119  0.087   -0.017
## 17 RI_Total_Product_Assortment c15+(a15*b11)+(a25*b12)  0.049  0.081   -0.006
## 18    RI_Total_Professionalism c18+(a18*b11)+(a28*b12)  0.105  0.078   -0.012
##    ci.upper
## 1    -0.001
## 2     0.003
## 3     0.046
## 4     0.090
## 5     0.050
## 6     0.056
## 7     0.153
## 8     0.085
## 9     0.077
## 10    0.008
## 11    0.003
## 12    0.269
## 13   -0.006
## 14    0.168
## 15   -0.036
## 16    0.255
## 17    0.104
## 18    0.221

Significant Coefficients at 5% alpha level

We will only account those coefficient and variables for this model.

parameterestimates(fit_SEM, standardized = TRUE) %>% select(lhs, rhs, est, pvalue,ci.lower, ci.upper) %>% filter(pvalue < 0.05) %>% filter(row_number() > 114) 
##        lhs     rhs    est pvalue ci.lower ci.upper
## 1  a12_b11 a12*b11 -0.021  0.036   -0.041   -0.001
## 2  a14_b11 a14*b11  0.024  0.032    0.002    0.046
## 3  a24_b12 a24*b12  0.059  0.000    0.029    0.090
## 4  a15_b11 a15*b11  0.030  0.003    0.010    0.050
## 5  a25_b12 a25*b12  0.034  0.003    0.011    0.056
## 6  a18_b11 a18*b11  0.097  0.001    0.042    0.153
## 7  a24_b22 a24*b22  0.177  0.000    0.086    0.269
## 8  a15_b21 a15*b21 -0.053  0.028   -0.100   -0.006
## 9  a25_b22 a25*b22  0.101  0.003    0.034    0.168
## 10 a18_b21 a18*b21 -0.171  0.013   -0.305   -0.036

Coefficient Interpretation

Indirect Effect

a12_b11: Store_Decoration -> Consumer_Satisfaction -> Repurchase_Intention (Negative)

a14_b11: French_Culture -> Consumer_Satisfaction -> Repurchase_Intention (Positive)

a24_b12: French_Culture -> Affective_Commitment -> Repurchase_Intention (Positive)

a15_b11: Product_Assortment -> Consumer_Satisfaction -> Repurchase_Intention (Positive)

a25_b12: Product_Assortment -> Affective_Commitment -> Repurchase_Intention (Positive)

a18_b11: Professionalism -> Consumer Satisfaction -> Repurchase_Intention (Positive)

a24_b12: French_Culture -> Affective_Commitment -> Repurchase_Intention (Positive)

a15_b21: Product_Assortment -> Consumer_Satisfaction -> Cocreation_Intention (Negative)

a25_b22: Product_Assortment -> Affective_Commitment -> Cocreation_Intention (Positive)

a18_b21: Professionalism -> Consumer_Satisfaction -> Cocreation_Intention (Negative)

SEM Visualization

Model without Covariances

lavaanPlot(model = fit_SEM, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "steelblue4"), coefs = TRUE, sig = 0.05, covs = FALSE, digits = 2)

Model with Covariances

lavaanPlot(model = fit_SEM, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "steelblue4"), coefs = TRUE,sig = 0.05, covs = TRUE, digits = 2)

Questions

1. What are the dimensions by which Galeries Lafayette is perceived? Please explain your findings and rational for your final result.

2. Are the mechanism driving satisfaction and affective commitment similar? Are satisfaction and affective commitment mediating the impact of image perceptions on outcomes? If yes for which outcomes?

3. What is driving the two distinct outcomes (repurchase and co-creation intention): Please rank the image dimensions with respect to the total effect on each outcome? Interpret your results.